The radius of a circle of area 112 sq. inches, after rounding off, is approximately equal to 6 inches.
We use the general formula for the area of a circle, which is defined as:
A = πr²
where 'r' is the radius of the circle.
Since we need to round off the decimal part, it is better to take π = 3.14 rather than 22/7, to avoid errors.
Also, note that the radius will be obtained in inches, corresponding to the units of area given.
By substituting the values given in the equation, we get:
112 = 3.14 * r²
r² = 112/3.14
r = √35.668
r = 5.97
After rounding off,
r ≅ 6 inches or 15 cm
Thus, the radius of the given circle is about 6 inches.
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In this problem, you will investigate the relationship between the area and perimeter of a rectangle.
b. Tabulate all possible whole-number values for the length and width of the rectangle, and find the area for each pair.
To investigate the relationship between the area and perimeter of a rectangle, we will tabulate all possible whole-number values for the length and width of the rectangle and find the area for each pair.
In a rectangle, the area is given by the formula A = length × width, and the perimeter is given by the formula P = 2(length + width). By systematically exploring different combinations of whole-number values for the length and width, we can calculate the corresponding area for each pair.
Table of Possible Whole-Number Values for Length and Width:
Length | Width | Area
-------|-------|-----
1 | 1 | 1
1 | 2 | 2
1 | 3 | 3
2 | 1 | 2
2 | 2 | 4
2 | 3 | 6
3 | 1 | 3
3 | 2 | 6
3 | 3 | 9
In the table above, we have listed all possible combinations of whole-number values for the length and width of the rectangle. For each combination, the corresponding area is calculated by multiplying the length and width.
By examining the table, we can observe that as the length and width increase, the area also increases. This demonstrates that there is a positive relationship between the area and the dimensions of the rectangle.
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angles x and y are supplementary. angle x is 3 times the measure of angle y. what is the measure of angle x? 45° 60° 120° 135°
Answer:
x = 135°
Step-by-step explanation:
x and y are supplementary angles, that is they sum to 180° , then
x + y = 180 ← substitute x = 3y into the equation
3y + y = 180
4y = 180 ( divide both sides by 4 )
y = 45
then
x = 3y = 3 × 45° = 135°
Simplify each expression.
√4 . 16
The solution of expression is, 32
We have to give that,
An expression to simplify,
⇒ √4 × 16
Now, We can take the square root of a number and simplify as.,
⇒ √4 × 16
⇒ 2 × 16
⇒ 32
Therefore, The solution is, 32
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Write the equation of the line parallel to 14x +7y =49
Answer:
14x + 7y = 49
2x + y = 7
y = -2x + 7
An equation of a parallel line is
y = -2x + c ---> 2x + y = c, where c is any constant. You can substitute any value for c.
What is the present value of a cash flow of $1500 if the rate of annual interest is 8.5 % ? Round to the nearest cent. The present value is
The present value of a cash flow of $1500 with an annual interest rate of 8.5% is approximately $1,062.74.
Present value (PV) is a financial concept used to determine the current worth of future cash flows, considering the time value of money. In this scenario, we can use the formula for calculating the present value of a single cash flow:
PV = CF / (1 + r)^n
Where PV is the present value, CF is the future cash flow, r is the annual interest rate (expressed as a decimal), and n is the number of periods (years in this case).
Now, let's calculate the present value of the $1500 cash flow with an 8.5% interest rate. We first convert the interest rate to a decimal: 8.5% = 0.085. Since the cash flow is received immediately (n = 0), the formula becomes:
PV = $1500 / (1 + 0.085)^0
PV = $1500 / 1
Therefore, the present value of the $1500 cash flow is $1500. This is because when the cash flow is received immediately, there is no compounding effect, and the present value is equal to the future cash flow amount. Thus, the present value is approximately $1,062.74 when rounded to the nearest cent, considering the time value of money at an 8.5% interest rate.
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In the diagram below, if the red lines are perpendicular bisectors, the point at which they cross is called the
Answer:
a. circumcenter
Step-by-step explanation:
You want to know the name of the point of intersection of the perpendicular bisectors of the sides of a triangle.
CircumcenterThe perpendicular bisectors of the sides of a triangle intersect at the "circumcenter." It is the center of a circle that circumscribes the triangle, intersecting all three vertices.
Effectively, each side of the triangle is a chord of the circumcircle. The perpendicular bisector of any chord passes through the center of the circle.
__
Additional comment
Other "centers" of a triangle are the centroid at the intersection of medians, the incenter at the intersection of angle bisectors, and the orthocenter at the intersection of altitudes.
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i roll five fair dice. i tell you at least two dice landed on 4, 5, or 6. what is the probability that there are exactly 4 dice that landed on a 4, 5, or 6
Using binomial probability, the probability of having exactly 4 dice on 4, 5 or 6 is 5/32.
What is the probability that there are exactly 4 dice that landed on a 4, 5, or 6?Using binomial probability, we can calculate the probability that out of the 5 dice thrown, the probability of having exactly 4 dice on 4, 5 or 6 can be calculated as;
[tex]P(X=k) = C(n, k) * p^k * (1-p)^(^n^-^k^)[/tex]
In the given data;
n = 5
k = 4
p = 3/6 = 1/2
Using the binomial probability formula, we can calculate:
P(X=4) = C(5, 4) * (1/2)⁴ * (1 - 1/2)⁵⁻⁴
P(X=4) = 5 * (1/2)⁴ * (1/2)¹
P(X=4) = 5 * (1/16) * (1/2)
P(X=4) = 5/32
Therefore, the probability that exactly 4 dice land on a 4, 5, or 6 when rolling five fair dice is 5/32.
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Write each polynomial in standard form. Then classify it by degree and by number of terms. 7x³ - 10x³ + x³.
Step-by-step explanation:
now,you can solve this question.
A canadian dealer bought 10 TVs from Japan for $250 each on 2011 . He subsequently sold 8 of them in 2011 for $450 each and the rest in 2012 for $40 each. By how much was the Canadian GDP affected in 2011 ? (if your answer is 100 dollars, for example, write 5100) Answer: $1020×($1600)
The Canadian GDP was affected by $1,632,000 in 2011.
To determine the impact on Canadian GDP, we need to consider the value added by the dealer through the sale of the TVs. GDP measures the total value of goods and services produced within a country's borders.
In 2011, the dealer bought 10 TVs from Japan for $250 each, resulting in a total expenditure of $2,500. The dealer sold 8 TVs in 2011 for $450 each, generating a revenue of $3,600. Therefore, the value added by the dealer in 2011 is the difference between the revenue from the sales and the initial expenditure, which is ($3,600 - $2,500) * 8 = $8,800.
However, the remaining 2 TVs were sold in 2012 for $40 each, which does not contribute to the GDP in 2011.
Therefore, the impact on Canadian GDP in 2011 due to the dealer's transactions is $8,800 * 100 = $880,000.
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Simplify each expression.
(x-3)(x-3)
The simplified expression for (x - 3)(x - 3) is x² - 6x + 9.
To simplify the expression (x - 3)(x - 3), we can apply the distributive property and then combine like terms:
(x - 3)(x - 3) = x(x) + x(-3) + (-3)(x) + (-3)(-3)
Using the distributive property:
= x² - 3x - 3x + 9
Combining like terms:
= x² - 6x + 9
Therefore, the simplified expression for (x - 3)(x - 3) is x² - 6x + 9.
The distributive property is a fundamental property in algebra that describes how multiplication distributes over addition or subtraction. It states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property allows us to simplify expressions by multiplying a value outside of a set of parentheses by each term inside the parentheses. The distributive property also holds true for subtraction:
a(b - c) = ab - ac
In both cases, the value outside the parentheses is distributed or applied to each term inside the parentheses individually. This property is particularly useful when dealing with expressions involving variables, as it allows us to simplify and manipulate expressions more easily.
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Solve each system.
y = x²+3 x+6
y = -x+2
The solution to the system of equations is x = -2 and y = 4.
To solve the system of equations:
Set the two equations equal to each other:
x² + 3x + 6 = -x + 2
Combine like terms and move all terms to one side to set the equation equal to zero:
x² + 4x + 4 = 0
Factor the quadratic equation:
(x + 2)(x + 2) = 0
Apply the zero-product property:
x + 2 = 0
Solve for x:
x = -2
Substitute the value of x back into either of the original equations to find the corresponding y-value:
y = (-(-2)) + 2
= 2 + 2
= 4
Therefore, the solution to the system of equations is x = -2 and y = 4.
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Look at the factors of 50 and 75.
Factors of 50: 1, 2, 5, 10, 25, 50
Factors of 75: 1, 3, 5, 15, 25, 75
The GCF of 50 and 75 is
Answer: Therefore, the GCF of 50 and 75 is 5.
Step-by-step explanation:
To find the greatest common factor (GCF) of 50 and 75, we can compare their factors.
Factors of 50: 1, 2, 5, 10, 25, 50
Factors of 75: 1, 3, 5, 15, 25, 75
By comparing the common factors between 50 and 75, we can see that the GCF is 5, as it is the largest number that divides both 50 and 75 without leaving a remainder.
Answer: 25
Step-by-step explanation:
explanation:
A bus travels 8.4 miles east
and then 14.7 miles north.
What is the magnitude of the
bus' resultant vector?
Hint: Draw a vector diagram.
[?] miles
Round your answer to the nearest hundredth.
Step-by-step explanation:
Using Pythagorean Theorem for right triangles
Resultant ^2 = 8.4^2 + 14.7^2
resultant = 16.93 miles
Recall the game of CHOMP from class (see Chapter 1 of Karlin+Peres). (a) How many possible positions might come up during the game starting with a 3×3 board? Include the starting and final position. (b) [bonus] How many possible position in the game started with an n×m board?
(a) In the game of CHOMP starting with a 3x3 board, there are a total of 14 possible positions that can arise, including the starting and final position.
(b) In general, for an n×m board, the number of possible positions in the game of CHOMP which is n×m.
(a) In the game of CHOMP, a position is defined by the configuration of the board, where each cell can be either "eaten" or "uneaten." Starting with a 3x3 board, there are a total of 9 cells. In each cell, the player can choose to either eat the cell or leave it uneaten. Since there are two possibilities (eaten or uneaten) for each cell, the total number of possible positions is [tex]2^9[/tex] = 512. However, not all of these positions are reachable during the game. Taking into account the rules of CHOMP, there are 14 distinct possible positions that can arise, including the starting and final position.
(b) For a general n×m board, the number of possible positions in the game of CHOMP can be determined by considering the number of cells on the board, which is n×m. In each cell, there are two possibilities (eaten or uneaten). Therefore, the total number of possible positions for an n×m board is [tex]2^(n×m)[/tex]. However, it is important to note that not all of these positions will be reachable during the game, as the reachable positions depend on the legal moves allowed in CHOMP.
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Solve by substitution.
y+5 x=-3
3 y-2 x=8
The solution to the system of equations is x = -1 and y = 2.
To solve the system of equations by substitution:
Start with the first equation:
y + 5x = -3 ...(Equation 1)
Solve Equation 1 for y:
y = -5x - 3
Substitute the value of y from Equation 1 into the second equation:
3y - 2x = 8
3(-5x - 3) - 2x = 8 ...(Substituting y = -5x - 3)
-15x - 9 - 2x = 8
-17x - 9 = 8
Solve the equation for x:
-17x = 8 + 9
-17x = 17
x = -1
Substitute the value of x into
y + 5(-1) = -3
y - 5 = -3
y = 2
Therefore, the solution to the system of equations is x = -1 and y = 2.
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identify the various measures of average and discuss the advantages and disadvantages of each. give examples of when one measure would be more useful than another.
The various measures of average is:
1. Arithmetic Mean:
2. Median
3. Mode
4. Geometric Mean
5. Harmonic Mean
1. Arithmetic Mean:
- Advantage: The arithmetic mean is the most widely used measure of average. It considers all data points and provides a balanced representation.
- Disadvantage: It is sensitive to extreme values (outliers) and can be influenced by skewed distributions.
- Example: Calculating the average height of a group of individuals.
2. Median:
- Advantage: The median is less affected by outliers and extreme values. It represents the middle value when the data is ordered.
- Disadvantage: It may not provide an accurate representation of the entire dataset, especially if the distribution is heavily skewed.
- Example: Determining the median income in a population to understand the typical earnings.
3. Mode:
- Advantage: The mode represents the most frequently occurring value(s) in the dataset. It is useful for identifying the most common category or value.
- Disadvantage: It may not exist or be unique in some datasets, or it may not provide a comprehensive summary of the data.
- Example: Identifying the most popular choice among a group of individuals.
4. Geometric Mean:
- Advantage: The geometric mean is useful when dealing with quantities that have multiplicative relationships, such as growth rates or compound interest.
- Disadvantage: It can only be calculated for positive numbers and is less commonly used for general data analysis.
- Example: Calculating the average annual growth rate of an investment portfolio.
5. Harmonic Mean:
- Advantage: The harmonic mean is appropriate for averaging rates, ratios, or speeds.
- Disadvantage: It is sensitive to extremely small values and may not be suitable for datasets with zero or negative values.
- Example: Determining the average speed of a trip when considering different segments.
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Write each function in vertex form.
f(x)= 4x²-8 x+2
The function f(x) = 4x² - 8x + 2 can be written in vertex form as f(x) = 4(x - 1)² - 2.
To convert the given function into vertex form, we need to complete the square. The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex.
Step 1: Group the first two terms and factor out the coefficient of x²:
f(x) = 4(x² - 2x) + 2.
Step 2: Complete the square by adding and subtracting the square of half the coefficient of x:
f(x) = 4(x² - 2x + 1 - 1) + 2.
Step 3: Factor the perfect square trinomial and simplify:
f(x) = 4((x - 1)² - 1) + 2.
Step 4: Distribute and combine like terms:
f(x) = 4(x - 1)² - 4 + 2.
Step 5: Simplify:
f(x) = 4(x - 1)² - 2.
Therefore, the given function f(x) = 4x² - 8x + 2 can be written in vertex form as f(x) = 4(x - 1)² - 2.
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A forest contains 24 elk, of which, 8 are captured, tagged, and released. a certain time later, 4 of the 24 elk are captured. what is the probability that 3 of these 4 have been tagged?
The probability that 3 out of the 4 captured elk have been tagged is approximately 0.0053.
To solve this problemWe can use the concept of combinations.
The total number of ways to choose 4 elk out of 24 is given by the combination formula:
C(24, 4) = 24! / (4!(24-4)!) = 10,626
Now, we need to consider the number of ways to choose 3 tagged elk out of the 8 tagged elk and 1 untagged elk. The number of ways to do this is given by:
C(8, 3) * C(1, 1) = 8! / (3!(8-3)!) * 1! / (1!(1-1)!) = 56
Therefore, the probability that 3 out of the 4 captured elk have been tagged is:
P = (Number of ways to choose 3 tagged elk out of 8 tagged elk and 1 untagged elk) / (Total number of ways to choose 4 elk out of 24)
P = 56 / 10,626
Calculating this division gives us the probability:
P ≈ 0.0053
So, the probability that 3 out of the 4 captured elk have been tagged is approximately 0.0053 .
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please guys I need all the help
Answer:
sin(θ) = 20 / 29
Step-by-step explanation:
Trigonometric ratios, or trig ratios for short, are mathematical ratios that relate the angles of a right triangle to the ratios of the lengths of its sides. These ratios are fundamental in trigonometry and are used to calculate various unknown angles or side lengths in a triangle.
The three primary trigonometric ratios are:
Sine (sin): The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse of the right triangle.
sin(θ) = (opposite side length) / (hypotenuse length)Cosine (cos): The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse of the right triangle.
cos(θ) = (adjacent side length) / (hypotenuse length)Tangent (tan): The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side of the right triangle.
tan(θ) = (opposite side length) / (adjacent side length)SOHCAHTOA is a mnemonic device used to remember the three primary trigonometric ratios in a right triangle: Sine, Cosine, and Tangent. It helps recall the relationships between these ratios and the sides of a right triangle.
Here's what each letter in SOHCAHTOA represents:
S = Sine
O = Opposite
H = Hypotenuse
C = Cosine
A = Adjacent
H = Hypotenuse
T = Tangent
O = Opposite
A = Adjacent[tex]\hrulefill[/tex]
Answering the question,
We are given a right triangle. The length of the hypotenuse with respect to theta is 29, the length of the opposite side with respect to theta is 20, and the length of the adjacent side with respect to theta is 21.
Recall: sin(θ) = (opposite side length) / (hypotenuse length)
Plug in what we know to find the trig ratio:
=> sin(θ) = 20 / 29
Thus, the sine trig ratio is found.
Film is fed through a traditional camera by gears that catch the perforation in the film. The distance from A to C is the same as the distance from B to D. Show that the two perforated strips are the same width.
It can be deduced that the two perforated strips in the film are of the same width.
Let's denote the width of the perforated strip (distance between two perforations) as "x."
The distance from A to C (AC) is equal to the distance from B to D (BD) since it's stated that both distances are the same.
This can be expressed as:
AC = BD
AC = AB (the width of the image, B to C) + BC (the width of the perforated strip, distance between two perforations)
BD = CD (the width of the image, B to D) + DC (the width of the perforated strip, distance between two perforations)
Since AC is equal to BD, we can set up an equation:
AB + BC = CD + DC
We know that BC and DC represent the width of the perforated strip, which is "x" in both cases.
So, we can rewrite the equation as:
AB + x = CD + x
Now, since AB is equal to CD (both represent the width of the image), we can further simplify the equation:
x = x
This equation shows that the width of the perforated strip is the same on both sides of the film.
Therefore, the two perforated strips are of the same width.
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Simplify each expression. Use only positive exponents. (-4 m²n³)(2 m n)
After simplification, the expression will become, -8.[tex]m^3n^4[/tex]
We know, [tex]a^x[/tex]×[tex]a^y[/tex]=[tex]a^{(x+y)[/tex]........ (i)
Where,
a ⇒ constant,
x and y⇒ different variables.
The given expression is,
(-4[tex]m^2n^3[/tex])(2mn) .
mn can be written as, [tex]m^1n^1[/tex].
Therefore, the above equation will be,
(-4[tex]m^2n^3[/tex])(2mn)
= (-4)×(2)×([tex]m^2n^3[/tex]×[tex]m^1n^1[/tex])
=(-8)×([tex]m^{2+1}n^{3+1[/tex])
=-8[tex]m^3n^4[/tex].
Hence, we got After simplifying using the positive exponents the expression will be, -8[tex]m^3n^4[/tex].
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Classify the following statement as true or false. If false, provide a counterexample.
If A,B,C,D, and E are collinear with B between A and C, C between B and D, and D between C and E, and A C=B D=C E , then A B=B C=D E .
The statement is false. A counterexample can be given to show its mistakenness.
Ponder the going with circumstance:
A- - - B- - - C- - - D- - - E
In this strategy, A, B, C, D, and E are collinear, with B among An and C, C among B and D, and D among C and E. Regardless, the statement ensures that A C = B D = C E, inducing that the distances between the centers are same.
In our counterexample, this condition isn't satisfied. By reviewing the strategy, clearly, A C isn't identical to B D, as part A C integrates both B and C, while B D simply integrates C. Also, B C isn't identical to D E since B C includes B and C, while D E integrates D and E.
Hence, the statement is false, as the counterexample shows what is going on where the value A C = B D = C E doesn't hold.
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Think About a Plan A cube-shaped jewelry box has a surface area of 300 square inches. What are the dimensions of the jewelry box?
(b) How is the side length of a square related to its area?
a. The cube-shaped jewelry box has a side length of approximately 7.071 inches. b. The side length of a square is related to its area through the formula: Area = side length^2.
a. To find the dimensions of the cube-shaped jewelry box, we need to determine the length of each side. Since a cube has all sides equal in length, we can find the side length by calculating the cube root of the surface area.
Let's denote the side length of the cube as "s". The formula for the surface area of a cube is given by:
Surface Area = 6 * s^2
The surface area is 300 square inches, we can set up the equation:
6 * s^2 = 300
Dividing both sides of the equation by 6, we get:
s^2 = 50
To solve for s, we can take the square root of both sides:
s = √50 ≈ 7.071
Therefore, the side length of the cube-shaped jewelry box is approximately 7.071 inches.
(b) The side length of a square is related to its area through the formula:
Area = side length^2
In other words, the area of a square is equal to the square of its side length.
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Can you provide the solution for this exercise?
Let u(w) = −(b − w)c. What restrictions on w, b, and c are required to ensure that u(w) is strictly increasing and strictly concave? Show that under those restrictions, u(w) displays increasing absolute risk aversion.
under the restrictions that c is negative to ensure strict concavity, the utility function u(w) = -(b - w)c displays increasing absolute risk aversion.
To ensure that u(w) is strictly increasing, we need the derivative of u(w) with respect to w to be positive for all values of w. Taking the derivative, we have du(w)/dw = -c. For u(w) to be strictly increasing, -c must be positive, which implies c must be negative.
To ensure that u(w) is strictly concave, we need the second derivative of u(w) with respect to w to be negative for all values of w. Taking the second derivative, we have d²u(w)/dw² = 0. Since the second derivative is constant and negative, u(w) is strictly concave.
Now, let's examine the concept of increasing absolute risk aversion. If a utility function u(w) exhibits increasing absolute risk aversion, it means that as wealth (w) increases, the individual becomes more risk-averse.
In the given utility function u(w) = -(b - w)c, when c is negative (as required for strict concavity), the absolute risk aversion increases as wealth (w) increases. This is because the negative sign implies that the utility function is concave, indicating that the individual becomes more risk-averse as wealth increases.
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Which scale would produce the largest scale drawing of an object when compared to the actual object?
The scale with the largest numerical value would produce the largest scale drawing.
When creating a scale drawing, we are representing an object or a structure on a smaller scale than its actual size. The scale is the ratio that relates the measurements of the drawing to the measurements of the actual object. It determines how much the drawing is reduced in size compared to the real object.
In scale drawings, a larger scale means that the drawing is closer in size to the actual object. The scale is usually expressed as a ratio, such as 1:100 or 1/4. The first number in the ratio represents the measurement on the drawing, while the second number represents the corresponding measurement on the actual object.
To determine which scale produces the largest scale drawing, we need to compare the numerical values of different scales. The larger the numerical value of the scale, the larger the drawing will be compared to the actual object. For example, a scale of 1:10 will result in a larger drawing than a scale of 1:100 because the first ratio has a larger numerical value. Similarly, a scale of 1/2 will produce a larger drawing than a scale of 1/4.
Therefore, the scale with the largest numerical value will produce the largest scale drawing of an object when compared to the actual object.
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Evaluate the following expression if x=2,y=-3, and z=1.
13+|8+y|
The value of the expression 13 + |8 + y| if x = 2, y = -3, and z = 1 is 18
How to evaluate the expressionFrom the question, we have the following parameters that can be used in our computation:
x = 2, y = -3, and z = 1
Also, we have
13 + |8 + y|
Substitute the known values in the above equation, so, we have the following representation
13 + |8 + y| = 13 + |8 - 3|
So, we have
13 + |8 + y| = 13 + |5|
Remove the absolute bracket
13 + |8 + y| = 13 + 5
So, we have
13 + |8 + y| = 18
Hence, the solution is 18
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Find the quotient.
5²/2
Answer:
12.5
Step-by-step explanation:
We are given:
[tex]\frac{5^2}{2}[/tex]
First, simplify by squaring 5:
[tex]\frac{25}{2}[/tex]
Then, divide to find your answer:
[tex]=12.5\\[/tex]
Hope this helps! :)
what is the value of x
Answer:
4.5
Step-by-step explanation:
The answer must be 4.5 because it is the only choice smaller than "6ft".
X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and m∠DAE = 65 , find the measure. D B
The measure of angle DBE is indeed 65°. This is a result of the properties of parallel lines and transversals in a rectangular fencing.
To explain further, we can use the properties of parallel lines and transversals. In the given figure, we have a rectangular fencing where AB and DE are parallel sides, and AD and BE are transversals.
Since AB and DE are parallel lines, the corresponding angles formed by the transversal AD are congruent. Therefore, we have:
m∠DAE = m∠DBE
Given that m∠DAE = 65°, we can conclude that m∠DBE is also 65° based on the congruence of corresponding angles.
Hence, the measure of angle DBE is indeed 65°. This is a result of the properties of parallel lines and transversals in a rectangular fencing.
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a rectangle is to be inscribed in an isosceles right triangle in such a way that one vertex of the rectangle is the intersection point of the legs of the triangle and the opposite vertex lies on the hypotenuse. find the largest area (in cm 2 ) of the rectangle and its dimensions (in cm) given that the two equal legs of the triangle have length 1.
To find the largest area of a rectangle inscribed in an isosceles right triangle with legs of length 1, we can determine the dimensions of the rectangle. The largest area is obtained when the rectangle's vertices touch the midpoint of the hypotenuse and the triangle's right angle vertex. The dimensions of the rectangle are \(1/2\) cm by \(1/2\) cm, resulting in an area of \(1/4\) cm\(^2\).
In an isosceles right triangle with legs of length 1, the hypotenuse has a length of \(\sqrt{2}\). The largest area of the inscribed rectangle occurs when its vertices touch the midpoint of the hypotenuse and the triangle's right angle vertex. This creates a rectangle with dimensions equal to half the lengths of the triangle's legs, resulting in a rectangle with dimensions \(1/2\) cm by \(1/2\) cm. The area of this rectangle is obtained by multiplying the lengths of its sides, which gives \(1/4\) cm\(^2\). Thus, the largest area of the rectangle is \(1/4\) cm\(^2\) with dimensions of \(1/2\) cm by \(1/2\) cm.
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