Using matrices, the system of equations is solved to find x = 118/17 and y = 2/17 as the solution. Matrix operations, including finding the inverse, are employed in the process.
To solve the given system of equations using matrices, we start by representing the system in matrix form.
The coefficient matrix A is obtained by arranging the coefficients of x and y, while the variable matrix X represents x and y as a column vector.
The constant matrix B contains the constants from the equations.
Next, we calculate the determinant of matrix A. If the determinant is nonzero, then A is invertible. In this case, the determinant of A is (2 * 1) - (-3 * 5) = 17, which is nonzero, so A is invertible.
To find the inverse of A, we proceed to calculate the cofactor matrix and then the adjugate matrix of A.
The adjugate matrix is the transpose of the cofactor matrix. By applying the formula A^(-1) = (1/det(A)) * Adj(A), we obtain the inverse matrix A^(-1).
Finally, we find the solution by multiplying A^(-1) by the constant matrix B. The product A^(-1) * B gives us the variable matrix X, which contains the values of x and y. In this case, the calculation yields x = 118/17 and y = 2/17.
Therefore, the solution to the system of equations is x = 118/17 and y = 2/17.
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Simplify each radical expression._ 1/√81
The simplified form of radical expression, 1/√81 is 1/9.
The given expression is,
1/√81
Simplify the denominator of the fraction 1/√81.
We know that the square root of 81 is 9 (because 9² = 81),
So √81 = 9.
Thus, the expression 1/√81 can be rewritten as 1/9.
Since,
9 = 3²
or
9 = 3⁵/3³
Therefore,
1/9 = 3³/3⁵.
Now, simplify this expression even further. 1/9 can be expressed as a fraction with a denominator of 3², or 3³/3⁵.
Therefore, we can write 1/√81 as 1/9 or 1/3², 3³/3⁵.
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Suppose that a function pairs elements from set A with elements from set B . A function is called onto if it pairs every element in B with at least one element in A . For each type of polynomial function, and for each set B , determine whether the function is always, sometimes, or never onto.
c. quadratic, B= all real numbers greater than or equal to 4.
The quadratic function is sometimes onto for the set B of all real numbers greater than or equal to 4, depending on the value of the leading coefficient a.
For a quadratic function, the general form is f(x) = ax^2 + bx + c, where a, b, and c are constants.
If we consider the set B to be all real numbers greater than or equal to 4, we need to determine whether the quadratic function can pair every element in B with at least one element in the set A (the domain of the function).
In this case, since the set B includes all real numbers greater than or equal to 4, we can find values of x for which the quadratic function will map to elements in B. Therefore, the quadratic function is sometimes onto for this set B.
To be more specific, for a quadratic function f(x) = ax^2 + bx + c, if the leading coefficient a is positive, the parabola opens upward, and there will be a minimum point. This minimum point will have a y-value greater than or equal to the minimum value of B. Therefore, the function can map elements from the domain A to the set B.
However, if the leading coefficient a is negative, the parabola opens downward, and there will be a maximum point. This maximum point will have a y-value less than or equal to the maximum value of B. In this case, the function will not be onto because there will be values of B that cannot be paired with any element in A.
Therefore, the quadratic function is sometimes onto for the set B of all real numbers greater than or equal to 4, depending on the value of the leading coefficient a.
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A touring boat was heading toward an island 80 nautical miles due south of where it left port. After traveling 15 nautical miles, it headed 8° east of south to avoid a fleet of commercial fishermen. After traveling 6 nautical miles, it turned to head directly toward the island. How far was the boat from the island at the time it turned?
b. What are you asked to find?
(a) the boat was 59 nautical miles away from the island when it turned.
(b) you are asked to find the distance between the boat and the island when it made the turn.
To answer question (a), we need to determine the distance between the boat and the island when it made the turn.
The boat initially traveled 15 nautical miles due south. Then, it changed its course by heading 8° east of south and traveled an additional 6 nautical miles. This forms a right triangle where the initial leg represents the 15 nautical miles due south, the additional leg represents the 6 nautical miles traveled after turning, and the hypotenuse represents the distance between the boat and the island.
To find the distance between the boat and the island when it turned, we can use trigonometry. We have a right triangle with one angle measuring 8° and one leg measuring 15 nautical miles. Since we want to find the hypotenuse, we can use the sine function:
sin(8°) = Opposite / Hypotenuse
Rearranging the equation, we get:
Hypotenuse = Opposite / sin(8°)
Substituting the values, we have:
Hypotenuse = 6 nautical miles / sin(8°)
To answer question (b), you are asked to find the distance between the boat and the island when it made the turn.
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a) Find the value of 2x + y when x = 4 and y = 3
b) Find the value of a² + b when a = -2 and b = 5
You want to get a car loan for $20,000 and pay it off over 4 years. If the rate is 14% compounded monthly, how much would your payments be at the end of each month? (round your answer)
Multiple Choice
$675
$417
$610
$475
$547
Rounding the answer to the nearest dollar, the monthly payment would be $475. Therefore, the correct answer is $475.
To calculate the monthly payments for a car loan, we can use the formula for the monthly payment on an amortizing loan:
P = (r * PV) / (1 - [tex](1 + r)^-^n[/tex])
Where:
P is the monthly payment
r is the monthly interest rate
PV is the loan principal (the amount borrowed)
n is the total number of payments
In this case, the loan principal (PV) is $20,000, the interest rate (r) is 14% per year, compounded monthly (monthly interest rate is 14%/12), and the total number of payments (n) is 4 years * 12 months/year = 48 months.
Let's calculate the monthly payment:
r = 14% / 12 = 0.14 / 12 = 0.01167 (monthly interest rate)
PV = $20,000
n = 48
P = (0.01167 * 20000) / (1 - [tex](1 + 0.01167)^-^4^8[/tex])
P ≈ $475.24
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Find the product. (y²)⁵ · y⁸
Answer: y^18
Step-by-step explanation:
(y²)⁵ · y⁸
Using exponent rules, you would multiply the exponents, 2 * 5 to get y^10
Now you have y^10 * y^8
This is another rule, you will add 10+8 because they are exponents
This leads to our answer y to the power of 18
Hope this helped
A company's weekly revenue R is given by the formula R=-p²+30 p , where p is the price of the company's product. The company is considering hiring a distributor, which will cost the company 4 p+25 per week.
b. Which value of p will maximize the profit after including the distributor cost?
The value of p that will maximize the profit after including the distributor cost is p = 13 where p is the price of the company's product by using profit function
The revenue function is given by [tex]R = -p^2 + 30p[/tex], and the distributor cost is [tex]4p + 25.[/tex]
To determine the value of p that will maximize the profit after including the distributor cost, is required to find the price that maximizes the company's revenue and subtract the distributor cost from it.
The profit function (P) can be calculated by subtracting the distributor cost from the revenue:
[tex]P = R - (4p + 25)\\= -p^2 + 30p - (4p + 25)\\= -p^2 + 30p - 4p - 25\\= -p^2 + 26p - 25[/tex]
To find the value of p that maximizes the profit, we need to find the vertex of the parabolic profit function, since the vertex represents the maximum point.
The vertex of a quadratic function in the form ax² + bx + c can be found using the formula:
[tex]p = -b / (2a)[/tex]
For the profit function [tex]-p^2 + 26p - 25[/tex].
Plugging these values [tex]a = -1, b = 26[/tex] into the formula, we get:
[tex]p = -26 / (2\times(-1))\\= -26 / (-2)\\= 13[/tex]
Therefore, the value of p that will maximize the profit after including the distributor cost is p = 13.
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Two positive angles that have a sum of /2 are ____________ angles, whereas two positive angles that have a sum of are __________ angles.
Two positive angles that have a sum of π/2 radians are complementary angles, whereas two positive angles that have a sum of π radians are supplementary angles.
Complementary angles are two angles whose measures add up to a right angle, which is equal to π/2 radians or 90 degrees. In other words, if α and β are complementary angles, then α + β = π/2.
Supplementary angles, on the other hand, are two angles whose measures add up to a straight angle, which is equal to π radians or 180 degrees. If α and β are supplementary angles, then α + β = π.
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Why don't the expressions 3+5² . 3/ 15 and (3 + 5²) . 3 / 15 yield the same answer?
The expressions 3+5² . 3/15 and (3 + 5²) . 3/15 do not yield the same answer due to the difference in the order of operations.
In mathematics, the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), determines the sequence in which mathematical operations are performed.
In the first expression, 3+5² . 3/15, the exponentiation (5²) is performed first, then the multiplication (5² . 3), and finally the division (5² . 3/15). This simplifies to 5² . 3/15 = 25 . 3/15 = 5.
In the second expression, (3 + 5²) . 3/15, the parentheses (3 + 5²) indicate that the addition (3 + 5²) is performed first, followed by the multiplication ((3 + 5²) . 3), and then the division (((3 + 5²) . 3) / 15).
This simplifies to (3 + 5²) . 3/15 = (3 + 25) . 3/15 = 28 . 3/15 = 84/15 = 5.6.
Thus, due to the difference in the order of operations, the two expressions yield different results.
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Suppose you want to fill nine 1-lb tins with a snack mix. You plan to buy almonds for 2.45/lb , peanuts for 1.85/lb , and raisins for .80 /lb . You want the mix to contain twice as much nuts as raisins by weight. If you spend exactly 15 , how much of each ingredient should you bu
b. How can you represent this system using a matrix equation?
To fill nine 1-lb tins with a snack mix containing twice as much nuts as raisins by weight using almonds for $2.45/lb. , peanuts for 1.85/lb., and raisins for 80 /lb. , you should buy 1.58 pounds of almonds, 3.16 pounds of peanuts, and 4.74 pounds of raisins.
You can use the following method to fill nine 1-lb tins with a snack mix:
Let x be the quantity of almonds in pounds.
Then 2x is the quantity of peanuts.
And 3x is the quantity of raisins.
To have twice as much nuts as raisins by weight, you must have 6 pounds of nuts and 3 pounds of raisins.
The cost of almonds is $2.45/lb, peanuts are 1.85/lb,andraisinsare.80/lb.
We can set up a system of equations to represent this problem:
2.45x + 1.85(2x) + .80(3x) = 15
Simplifying this equation gives:
7.95x = 12.6
Therefore, x = 1.58.
So you should buy 1.58 pounds of almonds, 3.16 pounds of peanuts, and 4.74 pounds of raisins.
To represent this system using a matrix equation, we can use the following matrix:
[2.45 3.70 .80] [x] [15]
[2x]
[3x]
This can be written as Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix.
To fill nine 1-lb tins with a snack mix containing twice as much nuts as raisins by weight using almonds for $2.45/lb. , peanuts for 1.85/lb., and raisins for 80 /lb. , you should buy 1.58 pounds of almonds, 3.16 pounds of peanuts, and 4.74 pounds of raisins1. We can represent this system using a matrix equation Ax = b where A is the coefficient matrix [2.45 3.70 .80], x is the variable matrix [x;2x;3x], and b is the constant matrix [15]. Solving this equation gives us x = [1.58;3.16;4.74]
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Due to a storm, a pilot flying at an altitude of 528 feet has to land. If he has a horizontal distance of 2000 feet to land, at what angle of depression should he land?
The pilot should land at an angle of depression of 15.96 degrees.
The angle of depression is the angle between the horizontal line and the line of sight from the pilot's position to the landing point.
We have a right triangle formed by the pilot's altitude, the horizontal distance to land, and the line of sight.
The pilot's altitude (opposite side) is 528 feet, and the horizontal distance (adjacent side) is 2000 feet.
We can use the tangent function to find the angle of depression.
Tangent (θ) = Opposite / Adjacent
Let's calculate the angle of depression:
Tangent (θ) = 528 / 2000
θ = arctan(528 / 2000)
Using a calculator or trigonometric tables, we can find the arctan value:
θ = 15.96 degrees
Therefore, the pilot should land at an angle of depression of 15.96 degrees.
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Determine whether each system has a unique solution. If it has a unique solution, find it.
y=(2/3)x-3 y=-x+7]
The unique solution to the given system of equations is x = 6 and y = 1.
To determine whether the given system of equations has a unique solution, we can compare the slopes of the two lines. If the slopes are different, the lines will intersect at a single point, resulting in a unique solution.
Let's analyze the equations:
Equation 1: y = (2/3)x - 3
Equation 2: y = -x + 7
Comparing the coefficients of x in both equations, we see that the slopes are different. The slope of Equation 1 is 2/3, and the slope of Equation 2 is -1.
Since the slopes are different, the system has a unique solution.
To find this unique solution, we can set the two equations equal to each other and solve for x:
(2/3)x - 3 = -x + 7
First, let's simplify the equation:
(2/3)x + x = 7 + 3
(5/3)x = 10
To isolate x, multiply both sides by 3/5:
(5/3)(3/5)x = (10)(3/5)
x = 6
Now that we have the value of x, we can substitute it back into either of the original equations to find the corresponding value of y.
Using Equation 1:
y = (2/3)(6) - 3
y = 4 - 3
y = 1
Therefore, the unique solution to the given system of equations is x = 6 and y = 1.
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Write each measure in radians. Express the answer in terms of π and as a decimal rounded to the nearest hundredth.
-15°
15° in radians is equal to -π/12 or approximately -0.26 radians when rounded to the nearest hundredth.
To convert -15° to radians, we can use the formula: radians = degrees × π / 180.
Applying this formula to -15°, we have:
radians = -15 × π / 180
radians = -π / 12
Therefore, -15° in radians is equal to -π/12 or approximately -0.26 radians when rounded to the nearest hundredth.
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Calculate the amount of money that will accumulate if Leslie leaves the money in the bank for 3,7 , and 17 year(s). b. Suppose Leslie moves her money into an account that pays 6 percent or one that pays 8 percent. Rework part (a) using 6 percent and 8 percent. c. What conclusions can you draw about the relationship between interest rates, time, and future sums from the calculations you just did? a. After placing $9,000 in a savings account paying annual compound interest of 4 percent, the amount of money that will accumulate if Leslie leaves the money in the bank for 3 year(s) is $ (Round to the nearest cent.)
If Leslie leaves $9,000 in a savings account for 3 years with an annual compound interest rate of 4 percent, the amount of money that will accumulate is approximately $10,174.88.
To calculate the future value of Leslie's investment, we can use the compound interest formula:
Future Value = Principal Amount * (1 + Interest Rate)^Time
Given that the principal amount is $9,000, the interest rate is 4 percent (or 0.04), and the time is 3 years, we can substitute these values into the formula:
Future Value = $9,000 * (1 + 0.04)^3
Future Value ≈ $9,000 * 1.124864
Future Value ≈ $10,174.88
Therefore, if Leslie leaves $9,000 in the savings account for 3 years at an annual compound interest rate of 4 percent, the amount that will accumulate is approximately $10,174.88.
This calculation demonstrates the relationship between interest rates, time, and future sums. As the interest rate increases, the future value of the investment also increases. Similarly, as the time period increases, the future value of the investment grows. This shows that higher interest rates and longer time periods have a compounding effect on the accumulation of funds. It emphasizes the importance of considering both the interest rate and the length of time when making financial decisions, as they significantly impact the growth of an investment.
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Identify the hypothesis and conclusion of each conditional statement.
If a quadrilateral has four congruent sides, then it is a square.
Hypothesis statement: A quadrilateral has four congruent sides, then it is a square.
Conclusion statement: Four sides and four ANGLES then it is a SQUARE.
We have to give that,
The statement is,
A quadrilateral has four congruent sides, then it is a square.
Hence, we get;
Hypothesis statement:
Four sides and four ANGLES then it is a SQUARE.
Conclusion statement:
Therefore, The hypothesis and conclusion of each conditional statement are,
Hypothesis statement: A quadrilateral has four congruent
Conclusion statement: A quadrilateral has Four sides and four ANGLES t
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A consultant has made two statements about sampling: X
1. An entire set of data from which a sample can be selected for analysis a called a distribution
2. Statistical methods can be applied to a representative sample in both exploratory and confirmatory data analysis
Are these statement accurate?
A. Only state 1 is accurate
B. Only state 2 is accurate
C. Both state are accurate
D. Neither state is accurate
40. FreeFree plc manufacturers maintain computers and Ibrahim is a planner in its Quality Assurance Department.
In terms of Mintzberg's descreprion of organizational structure, to which of FreeFree plc's 'building blocks' does Inbrahim belong?
A. Support staff
B. Middle line
C. Operating core
D. Technostructure
The correct answer for the first question is B.
Only statement 2 is accurate. Statement 1 is incorrect because an entire set of data is typically referred to as a population, not a distribution. A distribution refers to the pattern or spread of data within a population or sample.
For the second question, Ibrahim, as a planner in the Quality Assurance Department of Free Free plc, would belong to the operating core building block. Mintzberg's organizational structure theory defines the operating core as the individuals directly involved in producing the organization's products or services. Since Ibrahim is part of the Quality Assurance Department, which is responsible for ensuring the quality of the products, he falls under the operating core category. The support staff typically includes administrative or auxiliary personnel, the middle line refers to managers responsible for coordinating and overseeing different departments, and the technostructure consists of individuals who specialize in analytical and support functions related to the organization's operations.
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Multiply each pair of conjugates. (5-√11)(5+√11)
The product of the given conjugates is 14.
Given is an expression (5-√11)(5+√11), we need to find the product of this conjugate pair,
To multiply the pair of conjugates (5 - √11)(5 + √11), we can use the difference of squares formula.
According to this formula, the product of a binomial and its conjugate is always a difference of squares.
The formula is: (a - b)(a + b) = a² - b²
In this case, a = 5 and b = √11. So we have:
(5 - √11)(5 + √11) = (5² - (√11)²)
Simplifying further:
= (25 - 11)
= 14
Therefore, the product of the given conjugates is 14.
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consider the value of t such that the area to the left of −|t|−|t| plus the area to the right of |t||t| equals 0.050.05.
The value of t that satisfies the given condition is approximately 1.36.
To explain further, let's break down the problem and solve it step by step. We are looking for a value of t such that the combined area to the left of −|t| and to the right of |t| equals 0.05.
First, let's consider the area to the left of −|t|. Since t is negative in this case, we can rewrite the expression as -t. The area to the left of -t is the same as the area to the right of t. We want the combined area of these two regions to equal 0.05.
Next, let's consider the area to the right of |t|. This area is represented by 1 - the area to the left of |t|. Since |t| can be positive or negative, we take the absolute value to ensure the area is positive.
To find the value of t, we can set up the equation:
0.05 = (1 - the area to the left of |t|) + the area to the right of |t|
Simplifying this equation, we have:
0.05 = (1 - the area to the left of |t|) + the area to the left of |t|
Since the area to the left of |t| is the same as the area to the right of -|t|, we can rewrite the equation as:
0.05 = (1 - the area to the right of -|t|) + the area to the right of -|t|
Now, we need to find the value of t that satisfies this equation. By solving this equation, we find that t is approximately 1.36.
In summary, the value of t that satisfies the condition is approximately 1.36, where the combined area to the left of −|t| and to the right of |t| equals 0.05. This solution is obtained by setting up and solving the equation based on the given conditions.
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Evaluate the determinant of each matrix.
[1 0 0 -1 2 3 4 -1 2]
The determinant of the matrix [1 0 0 -1 2 3 4 -1 2] is -24. The determinant of a 3x3 matrix can be calculated using the following formula:
det(A) = a11 * det(A23) - a12 * det(A13) + a13 * det(A12)
where A11, A12, A13 are the elements in the first row of the matrix, and A23, A13, A12 are the determinants of the matrices formed by deleting the first row and column, the first column, and the first row, respectively.
In this case, the determinant is calculated as follows:
det(A) = 1 * det(-1 2 4 -1 2) - 0 * det(2 3 4 2) + 0 * det(2 3 -1 2)
= 1 * (-1)(2) - 0 * (2)(4) + 0 * (3)(-1)
= -24
Therefore, the determinant of the matrix [1 0 0 -1 2 3 4 -1 2] is -24.
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44 friends evenly divided up an
n-slice pizza. One of the friends, Harris, ate
1
11 fewer slice than he received.
How many slices of pizza did Harris eat?
Answer:
(n/4)-1
Step-by-step explanation:
We are dividing the pizza, which has n slices by 4 friends in total.
This can be represented as:
n/4
One friend out of the 4, had 1 slice less than what he received.
This can be represented as:
(n/4)-1
Hope this helps! :)
In the derivation of the fundamental conservation law, what units do 1. (a) (b) show that each term in (1.2) has units [quantity]/[time].
In the derivation of the fundamental conservation law, each term in equation (1.2) has units of [quantity]/[time].
In the derivation of the fundamental conservation law, equation (1.2) represents the balance equation for a certain physical quantity. Each term in this equation must have consistent units to maintain dimensional consistency.
Since the units of time are typically represented as [time], each term in equation (1.2) must have units of [quantity]/[time] to ensure that the equation is dimensionally balanced.
This means that the rate of change of the physical quantity, represented by the time derivative, has units of [quantity]/[time].
Additionally, the other terms in the equation, such as the flux or source terms, must also have units of [quantity]/[time] to match the rate of change term.
By maintaining the same units for each term, equation (1.2) satisfies the requirement of dimensional consistency.
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What is the exponential smoothing forecast for period 5 assuming that alpha \( =0.4 \) ? (Keep 2 decimals in your answer)
The exponential smoothing forecast for period 5, assuming an alpha (α) value of 0.4, is determined by applying the exponential smoothing formula.
The exponential smoothing formula is given by:
Forecast for period t = α * Actual value for period t + (1 - α) * Forecast for period t-1
In this case, we need the forecast for period 5, so we will use the formula to calculate it based on the available data.
To calculate the exponential smoothing forecast for period 5, we use the given alpha (α) value of 0.4 and the available data for previous periods. The formula considers the actual value for period t and the forecast for the previous period (t-1).
Since we are calculating the forecast for period 5, we need the actual value for period 5 and the forecast for period 4. However, these values are not provided in the given information. Without the actual value for period 5 or the forecast for period 4, it is not possible to provide a specific numerical answer for the exponential smoothing forecast for period 5.
The exponential smoothing technique is commonly used for forecasting, where the alpha (α) value determines the weight given to the most recent observations. A higher alpha value places more weight on recent data, while a lower alpha value gives more weight to historical data. By adjusting the alpha value, the forecast can be adjusted to respond more quickly or more slowly to changes in the data.
To calculate the exponential smoothing forecast for period 5, it is essential to have the necessary data points. With the provided alpha (α) value of 0.4, the formula can be applied once the required data is available.
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Charlie and Kris ordered a 16-inch pizza and cut the pizza into 12 slices.
b. If Kris ate 2 pieces, what area of the pizza did she eat?
The area of the pizza that Kris (2 pieces) ate is 18.84 square inches.
Given that, Charlie and Kris ordered a 16-inch pizza and cut the pizza into 12 slices.
So, here diameter of the pizza is 12 inches.
Then, radius = 12/2 = 6 inches
We know that, the area of a circle is πr².
Now, area of a pizza = 3.14×6²
= 3.14×36
= 113.04 square inches
So, area of 1 piece = 113.04/12
= 9.42
Area of 2 pieces = 9.42×2
= 18.84 square inches
Therefore, the area of the pizza that Kris (2 pieces) ate is 18.84 square inches.
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what is the y intercept of y=7
Answer:
7, there is no slope so 7 is the y intercept
Step-by-step explanation:
The y-intercept is :
7
Work/explanation:
The given equation is in y = mx + b form, where m = slope and b = y intercept. This equation is called slope intercept form.
In our case, the equation is y = 7, which is the same thing as y = 0x + 7; this means that the slope is 0 and the y-intercept is 7.
Hence, b = 7.If mo and Alex shares some raisins in the ratio of 7:5, mo gets 14 more raisins and Alex. How many raisins do they share?
Answer:
84
Explanation:
Just by looking at the ratio, you can see Mo gets 2 more raisins than Alex for every 12 raisins they share. Since Mo gets 14 more raisins, that means they shared (12 × 7) raisins, since 14 ÷ 2 = 7.
[tex]12[/tex] × [tex]7 = 84[/tex]
So they shared 84 raisins together, Mo having 49 and Alex having 35.
A rectangle is constructed with its base on the diameter of a semicircle with radius 13 and with its two other vertices on the semicircle . what are the dimensions of the rectangle with maximum area ?
The maximum value of the function for the area of the rectangle, obtained from the function for the area of the rectangle, indicates that the dimensions of the rectangle are;
Base length = 13·√(2) units
Height = 13·√(2)/2 units
What is the maximum value of a function?The maximum value of a function is the point at which the value of the function is a maximum.
The diameter of the semicircle, D, obtained from the radius of the semicircle is; D = 2 × 13 = 26
The maximum area of the rectangle can be found as follows;
Let h represent the height of the rectangle, and let b represent the length of half the base length of the rectangle, we get;
h² + b² = 13²
Therefore, we get, the area of the rectangle, can be found as follows;
A = 2·b·h
Plugging in b² = 13² - h², therefore; b = √(13² - h²)
A = 2 × √(13² - h²) × h
dA/dh = d(2 × √(13² - h²) × h)/dh = (4·h² - 338)·(√(-(h² - 169))/(h² - 169)
Therefore, at the maximum area, we get; dA/dh = 0, therefore;
dA/dh = (4·h² - 338)·(√(-(h² - 169))/(h² - 169) = 0
(4·h² - 338)·(√(-(h² - 169)) = 0
(4·h² - 338) = 0
4·h² = 338
h² = 338/4 = 84.5
h = ±13·√(2)/2
b = √(13² - (13·√(2)/2)²) = √(84.5) = ±13·√(2)/2
The dimensions of the rectangle that produces that maximum area are;
Base length, 2 × b = 13·√(2), and the height = 13·√(2)/2
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∠3 and ∠4 form a linear pair. The measure of ∠3 is four more than three times the measure of ∠4 . Find the measure of each angle.
Let's denote the measure of ∠3 as x and the measure of ∠4 as y.
According to the given information:
∠3 and ∠4 form a linear pair, which means they are adjacent angles formed by two intersecting lines. In a linear pair, the sum of the measures of the angles is 180 degrees. So, we have:
x + y = 180
The measure of ∠3 is four more than three times the measure of ∠4. Mathematically, this can be expressed as:
x = 3y + 4
Now we can solve this system of equations to find the values of x and y.
From equation 2, we can rewrite it as:
x - 3y = 4 (equation 3)
To solve equations 1 and 3 simultaneously, we can use substitution or elimination method.
Let's use the elimination method by multiplying equation 3 by -1:
-x + 3y = -4 (equation 4)
Now we can add equations 1 and 4 to eliminate x:
(x + y) + (-x + 3y) = 180 + (-4)
4y = 176
y = 44
Substitute the value of y back into equation 2 to find x:
x = 3(44) + 4
x = 132 + 4
x = 136
Therefore, the measure of ∠3 is 136 degrees and the measure of ∠4 is 44 degrees.
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Homework: WEEK 3 ASSIGNMENT - CHAPTER 3 & 4
Question 9, P 4-4 (similar to)
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Part 1
You have a balance of
$5,000
on your credit card, which charges an interest rate of
1.5%
per month. Looking at yourbudget, you figure you can make the following payments. Will they be enough to pay off your credit card?
Month
1
2
3
4
5
6
7
8
Payment
$490
$550
$610
$670
$730
$790
$850
$910
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Part 1
(Select from the drop-down menus.)
The present value of your payments is
▼
smaller than
larger than
equal to
the amount of the loan, so you
▼
will not
will
be able to pay off the loan.
The present value of the payments is smaller than the amount of the loan, so you will not be able to pay off the loan.
The present value of the payments refers to the current value of the future payments, considering the time value of money. In this case, the payments are made over a period of 8 months, and each payment is listed. To determine if the payments will be enough to pay off the credit card, we need to calculate the present value of these payments and compare it to the initial loan amount of $5,000.
Since the interest rate on the credit card is 1.5% per month, we need to discount each payment back to its present value. By discounting the payments, we can see the equivalent value of these payments in today's dollars. If the present value of the payments is equal to or greater than $5,000, it would be enough to pay off the loan. However, if the present value is smaller than $5,000, it means the payments will not be sufficient to pay off the loan.
In this case, without specific information about the discount rate used to calculate the present value, we cannot make an exact determination. However, based on the given information, which states that the present value of the payments is smaller than the amount of the loan, it suggests that the payments will not be enough to pay off the credit card debt.
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In 2012, I had registered to run the California International Marathon. On race morning, it had been pouring down rain for days and there were still torrential rains, sustained 25mph winds, and areas of street flooding. Conversation with my spouse at 4 am race day: Him: It's going to be awful out there, surely you're not crazy enough to actually run in these conditions. Me: I have to go, I paid for it. a) Assuming that if I had not already paid the (large and non-refundable) entry fee, I would definitely not run in those conditions, is it rational for me to decide to run because I had paid a large entry fee? Explain. b) At the end of the race (which I ran) there was a beer tent where each runner could get one free beer. The rain had stopped, the weather was nice and the line for free beer was 25-30 people long. I get significant positive utility from having a beer after a really long run, so the benefit was positive and the price was zero. Yet I skipped the beer. How 'could this be rational (as defined by economists)?
a) It is not necessarily rational to decide to run in unfavorable conditions solely because a large entry fee has been paid. Rationality is determined by weighing the expected benefits and costs of an action, considering factors such as personal safety and enjoyment.
a) Rational decision-making involves assessing the expected benefits and costs of an action and choosing the option that maximizes personal utility. In the case of running in adverse weather conditions, factors such as personal safety, physical well-being, and enjoyment of the experience should be considered alongside the sunk cost of the entry fee. While the entry fee represents a financial investment, it should not be the sole determinant of the decision. If the expected negative consequences of running in dangerous conditions outweigh the benefits derived from participating, it would be rational to prioritize personal well-being over the sunk cost of the fee.
b) Skipping the free beer after the race can be rational if the perceived benefits of having the beer are outweighed by other factors, such as the time and effort required to wait in line or the desire to prioritize other post-run activities.
b) Rationality, as defined by economists, takes into account the subjective preferences and trade-offs individuals make based on their utility. While the free beer may have a positive utility for you after a long run, skipping it can still be rational. The decision depends on the relative value you assign to different activities or opportunities available to you at that moment. Factors such as the length of the line, time constraints, and alternative options for post-run relaxation or recovery could influence your decision. If the perceived benefits of waiting in line for the free beer are outweighed by other activities or priorities, it would be rational to skip it and allocate your time and effort elsewhere. Rationality in this context involves optimizing overall utility based on personal preferences and circumstances.
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Find all values of x in the interval [0, 2] that satisfy the equation. (enter your answers as a comma-separated list.) 2 sin(2x) = 2 cos(x)
The values of x that satisfy the equation 2 sin(2x) = 2 cos(x) in the interval [0, 2] are:
x = π/6, x = 5π/6, x = π/2, x = 3π/2.
To find all values of x in the interval [0, 2] that satisfy the equation 2 sin(2x) = 2 cos(x), we can use trigonometric identities to simplify and solve the equation step-by-step.
First, let's simplify the equation using the double angle identity for sine: sin(2x) = 2sin(x)cos(x). The equation becomes:
2(2sin(x)cos(x)) = 2cos(x)
Next, we can cancel out the 2 and the cos(x) on both sides of the equation:
2sin(x)cos(x) = cos(x)
Now, we have two cases to consider:
Case 1: cos(x) ≠ 0
In this case, we can divide both sides of the equation by cos(x):
2sin(x) = 1
Divide both sides of the equation by 2:
sin(x) = 1/2
The solution for this case is x = π/6 and x = 5π/6 within the interval [0, 2].
Case 2: cos(x) = 0
In this case, we have cos(x) on the left side of the equation and 0 on the right side. This means that x must be equal to π/2 or 3π/2 within the interval [0, 2].
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