Half way between 4/5 and 14/15 is 13/15.
To find the halfway point between 4/5 and 14/15, we need to calculate the average of the two fractions. Here's the process:
1. Make sure the fractions have a common denominator. In this case, the least common denominator (LCD) for 5 and 15 is 15.
2. Convert the fractions to equivalent fractions with the common denominator: 4/5 becomes 12/15 (multiply both numerator and denominator by 3), while 14/15 stays the same.
3. Add the two equivalent fractions together: 12/15 + 14/15 = 26/15.
4. Divide the sum by 2 to find the halfway point: (26/15) ÷ 2. To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number: 26/15 × 1/2 = 26/30.
5. Simplify the resulting fraction: 26/30 can be simplified by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2 in this case. Thus, 26 ÷ 2 = 13, and 30 ÷ 2 = 15. The simplified fraction is 13/15.
So, the halfway point between 4/5 and 14/15 in its simplest form is 13/15.
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You want your savings account to have a total of $23,000 in it within 5 years. If you invest your money in an account that pays 6.8% interest compounded continuously, how much money must you have in your account now?
You must have approximately $16,465.55 in your account now to achieve a balance of $23,000 in 5 years with a 6.8% interest rate compounded continuously.
To achieve a savings account balance of $23,000 in 5 years with an interest rate of 6.8% compounded continuously, you will need to use the formula for continuous compounding: A = P * e^(rt), where A is the future value, P is the principal amount (initial deposit), r is the interest rate, t is the time in years, and e is the base of the natural logarithm (approximately 2.71828).
In this case, A = $23,000, r = 0.068, and t = 5 years. You need to solve for P, the principal amount:
$23,000 = P * e^(0.068 * 5)
Now, you can solve for P:
P = $23,000 / e^(0.068 * 5)
P ≈ $16,465.55
So, you must have approximately $16,465.55 in your account now to achieve a balance of $23,000 in 5 years with a 6.8% interest rate compounded continuously.
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In ΔSTU, s = 360 cm, t = 110 cm and u=450 cm. Find the measure of ∠U to the nearest 10th of a degree.
The measure of angle U to the nearest tenth is 39.6°
What is cosine rule?The cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.
C² = a²+b²-2abcosC
450² = 360²+110²+2(110)(360)cosU
202500 = 129600+ 12100+ 79200cosU
202500 = 141700+79200cosU
79200cosU = 202500-141700
79200cosU = 60800
cos U = 60800/79200
cos U = 0.77
U = 39.6°( nearest tenth)
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Find the Differentials of
1) z = x^2 - xy^2 + 4y^5
2) f(x,y) = (3x-y)/(x+2y)
3) f(x,y) = xe^x3y
1) To find the differentials of z = x^2 - xy^2 + 4y^5, we can use the total differential formula:
dz = (∂z/∂x)dx + (∂z/∂y)dy
Taking the partial derivatives of z with respect to x and y:
∂z/∂x = 2x - y^2
∂z/∂y = -2xy + 20y^4
Substituting these into the total differential formula:
dz = (2x - y^2)dx + (-2xy + 20y^4)dy
2) To find the differentials of f(x,y) = (3x-y)/(x+2y), we can again use the total differential formula:
df = (∂f/∂x)dx + (∂f/∂y)dy
Taking the partial derivatives of f with respect to x and y:
∂f/∂x = (y-3)/(x+2y)^2
∂f/∂y = (3x-2y)/(x+2y)^2
Substituting these into the total differential formula:
df = [(y-3)/(x+2y)^2]dx + [(3x-2y)/(x+2y)^2]dy
3) To find the differentials of f(x,y) = xe^x3y, we can once again use the total differential formula:
df = (∂f/∂x)dx + (∂f/∂y)dy
Taking the partial derivatives of f with respect to x and y:
∂f/∂x = e^(x3y) + 3xye^(x3y)
∂f/∂y = 3x^2e^(x3y)
Substituting these into the total differential formula:
df = (e^(x3y) + 3xye^(x3y))dx + (3x^2e^(x3y))dy
Here are the results:
1) For z = x^2 - xy^2 + 4y^5, the partial derivatives are:
∂z/∂x = 2x - y^2
∂z/∂y = -2xy + 20y^4
2) For f(x,y) = (3x-y)/(x+2y), the partial derivatives are:
∂f/∂x = (3(x+2y) - 3(3x-y))/(x+2y)^2
∂f/∂y = (-1(x+2y) + (x+2y))/(x+2y)^2
3) For f(x,y) = xe^(x^3y), the partial derivatives are:
∂f/∂x = e^(x^3y) * (1 + 3x^2y)
∂f/∂y = xe^(x^3y) * x^3
These partial derivatives represent the differentials for each respective function.
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Ruben paints one coat on one wall that us 3 1/2 yards long by 9 feet tall. He then paints one coat on two part walks that are each 4 feet talk by 1 1/2 yards long. What was the total area he paintex?
Ruben painted a total area of [tex]130.5 square feet.[/tex]
To determine the total area that Ruben painted, we need to find the area
of each wall and then add them together. Since the dimensions of the
walls are given in different units (yards and feet), we will first need to
convert them to a common unit.The first wall is 3 1/2 yards long by 9 feet
tall, which is equivalent to 10 1/2 feet long by 9 feet tall (since 1 yard = 3
feet).
The area of this wall is:
[tex]10 1/2 feet * 9 feet = 94.5 square feet[/tex]
The second two walls are each 4 feet tall by 1 1/2 yards long, which is
equivalent to 4 feet tall by 4.5 feet long (since 1 yard = 3 feet).
The area of each of these walls is:
[tex]4 feet* 4.5 feet = 18 square feet[/tex]
Since Ruben painted one coat on each wall, the total area he painted is:
[tex]94.5 square feet + 2 * 18 square feet = 130.5 square feet[/tex]
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A tennis ball is dropped from a certain height. Its height in feet is given by h(t)=−16t^2 +14 where t represents the time in seconds after launch. What is the ball’s initial height?
The initial height of the ball after launch is 14ft.
What is vertical motion?A vertical motion is a motion due to gravity. This means the velocity and height will depend on the acceleration due to gravity.
The height of vertical motion is given as;
H = ut ± 1/2 gt²
where u is the initial velocity and t is the time to reach max height.
The height of a ball is given by;
h(t) = -16t²+14
where t represents the time in seconds after launch.
The initial height after launch is when t = 0
h(t) = -16(0)² +14
h(t) = 14ft
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x^3y^2-343y^5
factoring polynomials
The polynomial is factored to y²(x³ - 7³y³)
How to determine the expressionNote that polynomials are described as expressions that are made up of terms, variables, coefficients, factors and constants.
Also, they have a degree greater than one.
Index forms are also seen as forms used to represent values that are too large or small in more convenient forms.
From the information given, we have that;
x³y²-343y⁵
Now, find the cube value of 343, we have;
343 = 7³
Substitute the value
x³y²- 7³y⁵
Factorize the common terms, we have;
y²(x³ - 7³y³)
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[tex]CD= \left[\begin{array}{ccc}e1&e2\\e3&e4\\\end{array}\right][/tex]
the determinant of the matrix is e1e4-e3e2
What is the determinant of a matrix?The determinant of a matrix is a scalar value that is a function of the entries. It characterizes some properties of the matrix and the linear map represented by it. The determinant is nonzero if and only if the matrix is invertible and an isomorphism exists.
Determinants are only defined for square matrices and encode certain properties of the matrices.
The determinant of a matrix is defined by the difference betweern the product of the right diagonal to the the product of the left diagonal
From the given question. the determinant of the matrix is e1*e4 -e3-e2 = e1e4-e3e2
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Solve the equation and check your solution: -2(x + 2) = 5 - 2x
Answer:
I think the answer might be -4 = 3x.
Step-by-step explanation:
-2 times x + 2 = -4 and 5 - 2x = 3x so i think the answer is -4 = 3x. Also, you're welcome if this helps.
HELP ME PLSSSS ANYBODY OF ANY AGE I WILL LEAVE A GOOD REVIEW
Answer:
the third one
Step-by-step explanation:
Answer:
Option 3 is the correct answer
Step-by-step explanation:
The surface area of a prism is the area of the full net.
The area of the full net is the sum of the areas of each part
For the given net, there are three rectangles, and two triangles.
The area for rectangles and triangles are given by the following formulas:
[tex]A_{rectangle}=base*height[/tex]
[tex]A_{triangle}=\frac{1}{2}*base*height[/tex]
It is important to recognize that due to the fact that the 3-D shape is a prism, the two triangles are congruent, and have exactly the same dimensions and area.
Looking at the options:
Option 1 has three products added together. This would be the base time height of each of the three rectangles. It does not include the area for either of the triangles.
Option 2 does have an extra term in front with 3 numbers multiplied together. It most closely resembles 2 times the product of the base and height of the triangle, but recall the area for a triangle is one-half of the base times height (this may make more sense when looking at option 3). This over-calculates the area of the triangle, and then doubles that over-calculated area (to match the second triangle)
Option 3 has an extra term in front with the number 2 times a parenthesis with 3 terms. These three terms represent the "one-half" from the formula for the area of a triangle, and the base and height of the triangle. The 2 in front of the parentheses represents that there are two of those triangles, both with that area. This correctly calculates the area of the net, and thus, the surface area of the triangular prism.
Option 4 has an extra term in front, similar to option 3 which calculates the area of one triangle correctly, but fails to account for the area of the second triangle.
Option 3 is the correct answer.
The radius of a circle is 18 in. Find its area in terms of pi
Answer:
324π
Step-by-step explanation:
Area of circle = r² · π
r = 18 in
Find its area in terms of pi.
We Take
18² · π = 324π
So, the area of the circle is 324π.
Answer:
A = 324π
Step-by-step explanation:
A = πr²
A = π(18)²
A = π(324)
A = 324π
Help asap please!!!!
Ella rolls a die and then flips a coin. The sample space for this compound event is represented in the table (His heads and Tis talls). Complete the table and the sentence beneath it. Die 1 2 3 4 5 6 heads H-1 H-2 H-3 H-5 H-6 Coin tails T-1 T-3 T-4 T-5 The size of the sample space is
The sample space for Ella's compound event where she rolls a die and then flips a coin can be represented in the table below:
Die: 1 2 3 4 5 6
Coin: H-1 H-2 H-3 H-5 H-6 T-1 T-3 T-4 T-5
The size of the sample space is the total number of possible outcomes, which in this case is the number of rows in the table. We can see that there are 9 rows in the table, so the size of the sample space is 9.
To understand the sample space, we can imagine that each row in the table represents a possible outcome of the compound event. For example, the first row represents the outcome where Ella rolls a 1 on the die and gets heads on the coin. The second row represents the outcome where Ella rolls a 2 on the die and also gets heads on the coin, and so on.
Understanding the sample space is important in probability theory because it allows us to calculate the probability of specific events occurring. By knowing the size of the sample space and the number of favorable outcomes, we can determine the probability of an event happening.
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Find the exact length of the curve. 36y² = (x² – 4)³, 5 ≤ x ≤ 9, y ≥ 0 = 96.666
The exact length of the curve is 112/3(√3 + 1), or approximately 96.666.
To find the exact length of the curve, we can use the formula for arc length:
L = ∫a^b √(1 + [f'(x)]²) dx
where f(x) = (x² - 4)^(3/2)/6, 5 ≤ x ≤ 9.
First, we find f'(x):
f'(x) = 3x(x² - 4)^(1/2)/12 = x(x² - 4)^(1/2)/4
Then we substitute f'(x) into the formula for arc length:
L = ∫5^9 √(1 + [x(x² - 4)^(1/2)/4]²) dx
L = ∫5^9 √(1 + x²(x² - 4)/16) dx
L = ∫5^9 √(16 + 16x²(x² - 4)/16) dx
L = ∫5^9 √(16x² + x^4 - 4x²) dx
L = ∫5^9 √(x^4 + 12x²) dx
L = ∫5^9 x²√(x^2 + 12) dx
We can use the substitution u = x^2 + 12, which gives du/dx = 2x and dx = du/2x, to simplify the integral:
L = (1/2)∫37^93 √u du
L = (1/2) [(2/3)u^(3/2)]_37^93
L = (1/3)[(125 + 108√3) - (13 + 36√3)]
L = (1/3)(112√3 + 112)
L = 112/3(√3 + 1)
Therefore, the exact length of the curve is 112/3(√3 + 1), or approximately 96.666.
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a group conducting a survey randomly selects adults in a certain region. of the 2,500 adults selected, 1,684 are men.
assuming that men and women have an equal chance of being selected the probability of the adults being chosen this way
by chance is less than 0.01. interpret the results of this calculation
The probability of the adults being chosen this way by chance is less than 0.01 interprets that group is more likely to choose men over women
A group conducting a survey randomly selects adults in a certain region. Of the 2,500 adults selected, 1,684 are men. The men and women have an equal chance The result of the survey is significant at the 0.01 level which means that the probability of group selection being the result of chance is 0.01 or less because the event is least likely to happen the group is more likely to select men over women.
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In a survey of 85 people, every fifth person had a pierced ear. How many people had a pierced ear? A 0.5 × 85 B 85 × 15 C 5÷85 D 85-4/5 E 85 × 0.25
Answer:
B
Step-by-step explanation:
Every fifth people means one person from 5 people in total. So when we convert that into numbers, it becomes [tex]\frac{1}{5}[/tex].
And in total there are 85 people involved, so the answer is
[tex]85[/tex] × [tex]\frac{1}{5}[/tex]
Answer:
Step-by-step explanation:
correct asnswer b
This sont Use a calculator or program to compute the first 10 iterations of Newton's method for the given function and initial approximation, f(x) = 2 sin x + 3x + 3, Xo = 1.5 Complete the table. (Do not round until the final answer. Then found to six decimal places as needed) k k XX 1 6 2 7 3 8 4 9 5 10
given: function f(x) = 2sin(x) + 3x + 3 ,Xo=1.5
1. Compute the derivative of the function, f'(x).
2. Use the iterative formula: Xₖ₊₁ = Xₖ - f(Xₖ) / f'(Xₖ)
3. Repeat the process 10 times.
First, let's find the derivative of f(x):
f'(x) = 2cos(x) + 3
Now, use the iterative formula to compute the iterations:
X₁ = X₀ - f(X₀) / f'(X₀)
X₂ = X₁ - f(X₁) / f'(X₁)
...
X₁₀ = X₉ - f(X₉) / f'(X₉)
Remember to not round any values until the final answer, and then round to six decimal places. Since I cannot actually compute the iterations, I encourage you to use a calculator or program to find the values for each Xₖ using the provided formula.
If a scale dilates a two dimensional object by factors of 2/3 it means that?
If a scale dilates a two-dimensional object by a factor of 2/3, it means that the image of the object will be reduced by a factor of 2/3. In other words, the length and width of the image will be 2/3 of the length and width of the original object.
For instance, consider a rectangle with length L and width W. If we dilate this rectangle by a factor of 2/3, the new length and width of the rectangle will be (2/3)L and (2/3)W, respectively. The area of the new rectangle will be (2/3)L x (2/3)W = (4/9)LW, which is 4/9 of the original area. This means that the image is smaller than the original rectangle, and this type of dilation is called a reduction.
Dilations can be used in different applications of mathematics, such as geometry, trigonometry, and algebra. They are useful for changing the scale or size of an object in a proportional way, without altering its basic shape or characteristics.
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The base of a cone has a radius
of 6 centimeters. The cone is
7 centimeters tall. What is the volume
of the cone to the nearest tenth? Use 3. 14 for it.
A. 260 cm
C. 263. 8 cm3
B. 263. 7 cm
D. 264. 0 cm3
The volume of the cone to the nearest tenth is 263.8 cm^3.
What is the volume, rounded to the nearest tenth, of a cone with a radius of 6 centimeters and a height of 7 centimeters?To find the volume of the cone, we first need to use the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.
We are given that the radius is 6 centimeters and the height is 7 centimeters, so we can substitute these values into the formula.
The formula for the volume of a cone is V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.
Using the given values, we can plug them into the formula and solve:
V = (1/3)π(6 cm)^2(7 cm)
V ≈ 263.7 cm^3
Rounding this to the nearest tenth gives us the final answer of 263.8 cm^3, which is option (C).
Since 3 is less than 5, we round down, which means the answer is 263.8 cm^3, as shown in option (C).
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A newspaper for a large city launches a new advertising campaign focusing on the number of digital subscriptions. the equation s(t)=31,500(1.034)t approximates the number of digital subscriptions s as a function of t months after the launch of the advertising campaign. determine the statements that interpret the parameters of the function s(t).
"t" increases, and the number of digital subscriptions grows exponentially at a rate of 3.4% per month.
How to determine the statements that interpret the parameters of the function s(t).The function s(t) = 31,500(1.034)^t gives an approximation of the number of digital subscriptions s as a function of t months after the launch of the advertising campaign.
The parameters of the function s(t) are:
31,500: This is the initial number of digital subscriptions at t = 0, when the advertising campaign is launched.
1.034: This is the growth rate of the number of digital subscriptions per month. It represents the percentage increase in the number of subscriptions each month due to the advertising campaign. Specifically, each month the number of subscriptions is multiplied by 1.034, which is the same as increasing it by 3.4%.
t: This is the time in months after the launch of the advertising campaign. It is the independent variable of the function that determines the number of digital subscriptions at any given time t.
Statements interpreting the parameters of the function s(t) are:
The initial number of digital subscriptions at t = 0 is 31,500.
For every month after the launch of the advertising campaign, the number of digital subscriptions increases by 3.4%, or a factor of 1.034.
The parameter t represents the time in months after the launch of the advertising campaign. As t increases, the number of digital subscriptions grows exponentially at a rate of 3.4% per month.
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Finx, Inc., purchased a truck for $40,000. The truck is expected to be driven 15,000 miles per year over a five-year period and then sold for approximately $5,000.
Determine depreciation for the first year of the truck's useful life by the straight-line and units-of-output methods if the truck is actually driven 16,000 miles. (Round depreciation per mile for the units-of-output method to the nearest whole cent).
The depreciation for the first year of the truck's useful life is $7,467 by the straight-line method and $2,720 by the units-of-output method.
Straight-line method:Depreciation per year = (Cost - Salvage value) / Useful life
Depreciation per year = (40,000 - 5,000) / 5 = $7,000
Depreciation for the first year = (16,000 / 15,000) x $7,000 = $7,467
Units-of-output method:Depreciation per mile = (Cost - Salvage value) / Total miles expected to be driven
Depreciation per mile = (40,000 - 5,000) / (5 x 15,000) = $0.17/mile
Depreciation for the first year = 16,000 x $0.17 = $2,720
Therefore, the depreciation for the first year of the truck's useful life is $7,467 by the straight-line method and $2,720 by the units-of-output method.
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Use the known MacLaurin series to build a series for each of the following functions. Be sure to show each step (layer) in expanded form along the way. Write your final answer in proper summation notation
f(x) = (e^2x - 1 - 2x)/2x^2
To build a series for the given function f(x) = (e^(2x) - 1 - 2x)/2x^2, we can start by finding the MacLaurin series for e^(2x) and then manipulate it to obtain the desired series.
The MacLaurin series for e^(2x) is given by:
e^(2x) = Σ (2x)^n / n! for n = 0 to ∞
Expanding the series, we get:
e^(2x) = 1 + 2x + 2x^2/2! + 2^3x^3/3! + 2^4x^4/4! + ...
Now, we can substitute this back into the original function:
f(x) = (e^(2x) - 1 - 2x)/2x^2 = (1 + 2x + 2x^2/2! + 2^3x^3/3! + 2^4x^4/4! + ... - 1 - 2x) / 2x^2
Simplifying, we have:
f(x) = (2x^2/2! + 2^3x^3/3! + 2^4x^4/4! + ...) / 2x^2
Now, we can divide by 2x^2 to obtain the series for f(x):
f(x) = 1/2! + 2x/3! + 2^3x^2/4! + 2^4x^3/5! + ...
Finally, we can write the final answer in proper summation notation:
f(x) = Σ (2^(n-1)x^(n-2)) / n! for n = 2 to ∞
To begin, we can write f(x) as:
f(x) = (1/2x^2)[e^(2x) - 1 - 2x]
Next, we will use the Maclaurin series for e^x, which is:
e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...
Substituting 2x for x, we have:
e^(2x) = 1 + 2x + (4x^2)/2! + (8x^3)/3! + ...
Expanding the first two terms of the numerator in f(x), we have:
f(x) = (1/2x^2)[(1 + 2x + (4x^2)/2! + (8x^3)/3! + ...) - 1 - 2x]
Simplifying, we get:
f(x) = (1/2x^2)[2x + (4x^2)/2! + (8x^3)/3! + ...]
Now we can simplify the coefficients in the numerator by factoring out 2x:
f(x) = (1/x)[1 + (2x)/2! + (4x^2)/3! + ...]
Finally, we can write the series in summation notation:
f(x) = Σ[(2n)!/(2^n*n!)]x^n, n=1 to infinity.
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Jesse can mow 3 yards in 8 hours. Jackson can mow twice as many yards per hour. What is the constant of proportionality between the number of yards Jackson can mow and the number of hours?
If Jesse can mow 3 yards in 8 hours. Jackson can mow twice as many yards per hour the constant of proportionality between the number of yards Jackson can mow and the number of hours is 3/4.
To find the constant of proportionality between the number of yards Jackson can mow and the number of hours, we can use the formula:
k = y/x
where k is the constant of proportionality, y is the number of yards, and x is the number of hours.
We know that Jesse can mow 3 yards in 8 hours, which means his rate of mowing is: 3 yards/8 hours = 3/8 yards per hour
We also know that Jackson can mow twice as many yards per hour as Jesse, which means his rate of mowing is:
2 * (3/8) yards per hour = 3/4 yards per hour
Now we can use the formula to find the constant of proportionality for Jackson:
k = y/x = (3/4) yards per hour / 1 hour = 3/4
Therefore, the constant of proportionality between the number of yards Jackson can mow and the number of hours is 3/4.
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Can someone please help me with this
Can someone please help me ASAP? It’s due tomorrow. I will give brainliest if it’s correct. Show work.
The difference between the outcomes when selected with or without replacement is B. 10 outcomes.
How to find the number of outcomes ?With Replacement:
When you select two coins with replacement, you put the first coin back in the jar before selecting the second coin. This means that there are 10 possibilities for each selection. So, the number of outcomes for selecting two coins with replacement is 10 x 10 = 100 outcomes.
Without Replacement:
When you select two coins without replacement, you don't put the first coin back in the jar before selecting the second coin. This means that after selecting the first coin, there are 9 coins left in the jar for the second selection. So, the number of outcomes for selecting two coins without replacement is 10 x 9 = 90 outcomes.
Difference = 100 outcomes - 90 outcomes
Difference = 10 outcomes
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On a coordinate plane, kite g d e f has points (0, 0), (10, 10), (16, 8), (14, 2). complete the steps to find the area of the kite. what is ge? square root of units what is df? square root of units what is the area of the kite to the nearest unit? square unitson a coordinate plane, kite g d e f has points (0, 0), (10, 10), (16, 8), (14, 2). complete the steps to find the area of the kite. what is ge? square root of units what is df? square root of units what is the area of the kite to the nearest unit? square units
The square root of units is sqrt (80), and the area of the kite to the nearest unit is 74 square units.
To find the area of the kite, we can divide it into two triangles by drawing a diagonal between points (10,10) and (14,2).
First, we need to find the length of this diagonal. We can use the distance formula:
d = sqrt((14-10)^2 + (2-10)^2)
d = sqrt(16 + 64)
d = sqrt(80)
So the length of the diagonal is square root(80) units.
Next, we can find the area of each triangle:
Triangle 1:
Base = 10 units
Height = 10 units
Area = 1/2 * base * height = 1/2 * 10 * 10 = 50 square units
Triangle 2:
Base = 6 units (the difference between the x-coordinate plane of (10,10) and (14,2))
Height = 8 units (the difference between the y-coordinate plane of (10,10) and (16,8))
Area = 1/2 * base * height = 1/2 * 6 * 8 = 24 square units
So the total area of the kite is 50 + 24 = 74 square units.
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Answer:
On a coordinate plane, kite G D E F has points (0, 0), (10, 10), (16, 8), (14, 2).
Complete the steps to find the area of the kite.
What is GE?
Square root of
✔ 320units
What is DF?
Square root of
✔ 80units
What is the area of the kite to the nearest unit?
✔ 80
square units
The equation a² + b² = c² represents the relationship between the three sides of a right triangle.
Ivan is cutting a piece of fabric for his sewing project in the shape of a right triangle. His right triangle has a leg with
a length of 5 inches and a hypotenuse with a length of 11 inches. What is the length, in inches, of the other leg of
his triangle?
the length, in inches, of the other leg of his triangle is 9. 8inches
How to determine the lengthUsing the Pythagorean theorem which states that the square of the longest leg or side of a given triangle is equal to the sum of the squares of the other two sides of the triangle.
From the information given, we have that;
a² + b² = c² represents the relationship between the three sides of a right triangle
Also,
Hypotenuse side = 11 inches
One of the other side = 5 inches
Substitute the values, we have;
11² = 5² + c²
collect like terms
c² = 121 - 25
Subtract the values
c = √96
c = 9. 8 inches
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Your bank account consists of a checking and savings accounts. Assume your expenses and earnings can be described by a random walk with an equal probability to spend one dollar or to receive one dollar in your checking account at every time interval. You are charged $5 for any transaction from the checking account to the savings account and viceversa. Also, assume that the cost per unit of cash, per unit of time r of keeping cash on hand is equal to $0. 1 dollars for any dollar on hand per time period. Determine:
a. The optimal values of the two thresholds s and S, i. E. , the amount of cash in your checking account restored after each transaction, and the maximum amount of cash in your checking account, respectively.
b. The long run average cost associated to the optimal cash management strategy and to the strategy with the same s but with a maximum amount of cash equal to 2S.
c. Are there any common criticisms of this model?
a. To determine the optimal values of the two thresholds s and S, we can use the Miller-Orr cash management model. The objective is to minimize the total cost of cash management, which includes transaction costs and the opportunity cost of holding cash.
Let's assume that the transaction cost of $5 applies whenever the cash balance in the checking account goes below s or above S. The expected daily cash balance is zero since expenses and earnings are equally likely, and the standard deviation of the cash balance is σ = √(t/2), where t is the time interval.
The optimal value of s is given by:
s* = √(3rT/4C) - σ/2,
where T is the length of the cash management period, and C is the fixed cost per transaction. The optimal value of S is given by:
S* = 3s*,
which ensures that the probability of a cash balance exceeding S is less than 1/3.
Using r = 0.1, T = 1 day, and C = $5, we obtain:
s* = √(30.11/4*5) - √(1/2)/2 = $16.82
S* = 3*$16.82 = $50.47
Therefore, the optimal values of the two thresholds are s* = $16.82 and S* = $50.47.
b. The long run average cost associated with the optimal cash management strategy can be calculated as:
Total cost = (s*/2 + S*) * σ * √(2r/C) + C * E(N),
where E(N) is the expected number of transactions per day. Since expenses and earnings are equally likely, E(N) = (S* - s*)/2 = $16.83. Therefore, the total cost is:
Total cost = ($16.82/2 + $50.47) * √(1/2) * √(2*0.1/$5) + $5 * $16.83 = $1.38 per day.
Now let's consider the strategy with the same s but with a maximum amount of cash equal to 2S. The expected daily cash balance is still zero, but the standard deviation is now σ' = √(t/3). The optimal value of S' is given by:
S' = √(3rT/2C) - σ'/2 = $35.35.
The long run average cost associated with this strategy is:
Total cost' = (s/2 + S') * σ' * √(2r/C) + C * E(N'),
where E(N') is the expected number of transactions per day. Since the maximum amount of cash is now 2S, we have E(N') = (2S - s)/2 = $34.59. Therefore, the total cost is:
Total cost' = ($16.82/2 + $35.35) * √(1/3) * √(2*0.1/$5) + $5 * $34.59 = $1.30 per day.
Therefore, the strategy with the same s but with a maximum amount of cash equal to 2S is slightly more cost-effective in the long run.
c. One common criticism of this model is that it assumes a constant transaction cost, which may not be realistic in practice. In reality, transaction costs may vary depending on the size and frequency of transactions, and may also depend on the banking institution and the type of account. Another criticism is that it assumes a random walk model for expenses and earnings, which may not capture the
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If a person drives his car at the speed of 50 miles per hour, how far can he cover in 2.5 hours?
An architect draws a blueprint of the newly modeled family room she is designing for her basement. The scale she uses is 1 inch = 2.5 feet. If the length of the family room is 8 inches, and the width of the family room is 4 inches, what are the actual dimensions of the family room?
Answer:20 feet by 10 feet
Step-by-step explanation:
As President of Spirit Club, Rachel organized a "Day of Decades" fundraiser where students could pay a fixed amount to dress up as their favorite decade. Of the 19 students who participated, 15 of them dressed up as the '40s.
If Rachel randomly chose 16 of the participants to take pictures of for the yearbook, what is the probability that exactly 13 of the chosen students dressed up as the '40s?
Write your answer as a decimal rounded to four decimal places.
The probability of choosing exactly 13 students who dressed up as the 40s out of the 16 selected students is approx. 0.4334.
What is the probability of choosing exactly 13 students who dressed up as the 40s?We can model this situation as a hypergeometric distribution, where we have a population of 19 students, 15 of whom dressed up as the 40s.
We want to choose a sample of 16 students and find the probability that exactly 13 of them dressed up as the '40s.
The probability of choosing exactly 13 students who dressed up as the 40s can be calculated:
(number of ways to choose 13 students who dressed up as the 40s) * (number of ways to choose 3 students who dressed up as other decades) / (total number of ways to choose 16 students)
The number of ways to choose 13 students who dressed up as the '40s is the number of combinations of 15 choose 13:
(15 choose 13) = 105
The number of ways to choose 3 students who dressed up as other decades is the number of combinations of 4 choose 3, which is:
(4 choose 3) = 4
The total number of ways to choose 16 students from 19 is the number of combinations of 19 choose 16, which is:
(19 choose 16) = 969
105 * 4 / 969 = 0.4334
Therefore, the probability of choosing exactly 13 students who dressed up as the 40s = 0.4334 (rounded to four decimal places)
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Match each phrase with the type of inequality it indicates.
The inequalities represented are:
Below - Less than or equal toAbove - Greater thanMore than - Greater thanSmaller Than - Less thanAt most - Less than or equal toAt least - Greater than or equal toNo more than - Less than or equal toNo less than - Greater than or equal toNot to exceed - Less than or equal toMaximum - Less than or equal toWhat is an inequality?In mathematics, an inequality is a statement that two values or expressions are not equal. It is used to compare two values and determine the relationship between them. Inequalities use symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
Inequalities can be solved and graphed on a number line to show all possible solutions that satisfy the inequality. They are commonly used in algebra and calculus to express a range of values for a variable.
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