Rewrite each expression using a single, positive exponent.

The equation that has given solutions x = -5 and x = 2 is [tex]x^2 + 3x - 10 = 0.[/tex]

If the given solutions of an equation are x = -5 and x = 2, then the equation can be written as a product of two linear factors, (x + 5) and (x - 2), because when either of these factors is equal to zero, the corresponding solution is obtained.

So, the equation is:

(x + 5)(x - 2) = 0

Expanding the product, we get:

[tex]x^2 + 3x - 10 = 0[/tex]

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Answer:

x=137

Step-by-step explanation:

sqrt(x+7)=12

x+7=144

x=137

You lose 515 dollars every time you lose 10 dollars.

Now that we know it takes 50 hours to lose 10 dollars, we can calculate how much money you lose every 10 dollars by

the hourly rate of 10.50 dollars by 50 hours and subtracting that amount from 10 dollars:

Money lost = (10.50 dollars/hour) x 50 hours - 10 dollars

Money lost = 525 dollars - 10 dollars

Money lost = 515 dollars

Therefore, you lose 515 dollars every time you lose 10 dollars.

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The median for the number of pencils can be found by locating the middle value on the dot plot, which appears to be 3 pencils.

The dot plot displays the number of pencils each boy has at his desk in class. By locating the middle value of the data on the dot plot, the median number of pencils can be determined to be 3.

This indicates that half of the boys have 3 pencils or fewer, while the other half have 4 pencils or more.

The dot plot provides a visual representation of the distribution of the data and allows for easy identification of the median value.

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The area of this irregular polygon is 344.5 square units.

To find the area of an irregular polygon, you can divide it into smaller, simpler shapes.

In this case, we can divide the polygon into a rectangle and a right triangle.

The rectangle has a height of 4 and a width of 13, so its area is 4 x 13 = 52 square units.

The right triangle has a base of 13 and a height of (110 - 4 x 13) / 2 = 45, since the total height of the polygon is 110.

Therefore, the area of the right triangle is (1/2) x base x height = (1/2) x 13 x 45 = 292.5 square units.

Adding the areas of the rectangle and the triangle, we get a total area of 52 + 292.5 = 344.5 square units.

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When comparing the values of f(4) and g(4), we need to take into account the fact that f(x) is exponential and g(x) is linear. Exponential functions grow at an increasing rate as x increases, while linear functions grow at a constant rate. Therefore, as x gets larger, the difference between f(x) and g(x) will become greater.

To find out how much greater f(4) is than g(4), we first need to calculate the values of f(4) and g(4). Let's say that f(x) = 2^x and g(x) = 3x + 1. Plugging in x = 4, we get:
f(4) = 2^4 = 16
g(4) = 3(4) + 1 = 13

So, f(4) is greater than g(4) by a difference of 3. However, this does not take into account the fact that f(x) is exponential and g(x) is linear.

To see the impact of the different growth rates, let's compare the values of f(x) and g(x) for a range of values of x. We can create a table to compare the two functions:

x    f(x)    g(x)
0    1       1
1    2       4
2    4       7
3    8       10
4    16      13
5    32      16

From this table, we can see that as x increases, the difference between f(x) and g(x) grows at an increasing rate. This is because f(x) is growing exponentially, while g(x) is growing linearly.

In summary, f(4) is 16 and g(4) is 13, so f(4) is greater than g(4) by a difference of 3. However, we also need to take into account the fact that f(x) is exponential and g(x) is linear. As x increases, the difference between f(x) and g(x) will grow at an increasing rate. Therefore, the difference between f(4) and g(4) is not only 3, but also growing exponentially.

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Therefore , the solution of the given problem of function comes out to be Line B  to the function y = x+5.

On the midterm exam, there will be a variety of inquiries in each subject, including inquiries concerning both hypothetical and actual places as well as inquiries relating the creation of numerical variables. a diagram showing the relationships between different elements that cooperate to generate the same result. A service is composed of numerous distinctive components that cooperate to create distinctive results for each input.

Here,

The function is represented as

=> y = x + 5, which has a y-intercept of 5, a slope of 1, and a slope of 1.

By comparing the slope and y-intercept of each line to those of the function, we may determine which line best fits it.

Line A does not fit the function because it has a slope of 1 and a y-intercept of 0.

The function y = x + 5 is satisfied by line B, which has a slope of 1 and a y-intercept of 5.

Line C does not fit the function because it has a slope of 2 and a y-intercept of 0.

Line D does not fit the function because it has a slope of 3 and a y-intercept of 2.

Line B therefore corresponds to the function y = x+5.

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Answer: D)-3/4

Step-by-step explanation: Rate of change of y with respect to x implies the following formula...(y2-y1)/(x2-x1)...which is also the gradient of the line, you can substitute the value in accordance with the points already ahown in the grid to get, (7-1)/(-4-4)=-3/4

Answer:

IT depends on how many kids there are per group.

Step-by-step explanation:

The answer is 37.5.

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The correct option is (a)  270 degree clockwise rotation because they both result in the same final orientation, just in opposite directions.

A 90-degree counter-clockwise rotational  symmetry  is the same as a 270-degree clockwise rotation. This is because rotations are measured in degrees, and a full circle is 360 degrees. Therefore, rotating something 90 degrees counter-clockwise means turning it a quarter of a full circle, while a 270-degree clockwise rotation means turning it three-quarters of a full circle.

To visualize this, imagine a square with its sides parallel to the x and y axes. If we rotate this square 90 degrees counter-clockwise, its sides will now be parallel to the y and negative x axes.

However, if we rotate the same square 270 degrees clockwise, its sides will also be parallel to the y and negative x axes, as it has been turned three-quarters of the full circle in the opposite direction.

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There are 252 different combinations of shirts, the correct option is B.

Remember that if you have N elements, and you want to select K of these, the number of combinations is given by:

[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]

Here we have 10 shirts and we want to select 5 of these, so the number of combinations is given by:

[tex]C(10, 5) = \frac{10!}{(10 - 5)!*5!} \\\\C(10, 5) = \frac{10*9*8*7*6}{5*4*3*2} = 252[/tex]

There are 252 different combinations, so the correct option is B.

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I apologize, but there seems to be some confusion in your question. The function given, y = ∫√25sin^2t - 1 dt, is not a curve but rather an indefinite integral expression. In order to find the length of a curve, we need a function defined explicitly in terms of x (or y) and its bounds. Could you please provide more information or clarify your question?
To find the length of the curve given by y = ∫√(25sin^2(t) - 1) dt from 0 to π/2, we need to calculate the definite integral.

First, let's set up the integral:

Length = ∫√(25sin^2(t) - 1) dt, with bounds from 0 to π/2

Unfortunately, this integral cannot be solved analytically using elementary functions. You will need to use a numerical method, such as the Trapezoidal Rule or Simpson's Rule, to approximate the value of the integral, and thus find the length of the curve.

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Answer:

Part (a):  Because the survey was held at basketball championship.

Part (b): The survey among students should held at school or in any common place.

Step-by-step explanation:

Place value is the value of a digit based on its position within a number system. To solve the equation 19 + 50, we use place value to add the digits in the ones and tens place.

Place value is a fundamental concept in mathematics that helps in understanding the value of digits in a number system. In the decimal system, each digit's value depends on its position relative to the decimal point, with the position to the left representing higher values than those to the right.

To solve 19 + 50, we add the digits in the ones place (9 + 0 = 9) and the digits in the tens place (1 + 5 = 6) separately, then combine the results to get the final answer of 69. This is possible due to the place value concept. The digit 9 in the ones place of 19 represents a value of 9 units, while the digit 1 in the tens place represents 10 units.

Similarly, the digit 5 in the tens place of 50 represents 50 units, and the digit 0 in the ones place represents 0 units. By adding the digits based on their place value, we get the answer 69, where 9 is in the ones place and 6 is in the tens place.

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The surface area of the hexagonal pyramid is 75240 square millimeters to the nearest square millimeter.

The surface area of the hexagonal pyramid can be calculated by finding the sum of the areas of its faces. The hexagonal pyramid has a base with six equal sides of length 44 millimeters, and six identical triangular faces with a height of 95 millimeters.

Each triangular face is an isosceles triangle, with two sides of length 44 millimeters and a base of length equal to the perimeter of the hexagonal base, which is 6 times 44 millimeters, or 264 millimeters.

To calculate the area of each triangular face, we can use the formula for the area of an isosceles triangle, which is (base x height) / 2. Substituting the values we have, we get: Area of each triangular face = (264 x 95) / 2 = 12540 square millimeters

Since the hexagonal pyramid has six identical triangular faces, we can multiply the area of one triangular face by 6 to get the total surface area of the pyramid: Total surface area = 6 x 12540 = 75240 square millimeters.

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It should be noted that to calculate the mean and standard deviation of the sampling distribution, you need to know the population mean (μ) and standard deviation (σ) and the sample size (n) of the distribution.

The mean (μx) of the sampling distribution of the sample mean (x) is equal to the population mean (μ):

μx = μ

The standard deviation (σx) of the sampling distribution of the sample mean (x is equal to the population standard deviation (σ) divided by the square root of the sample size (n):

σx = σ / √n

Therefore, in order to calculate the mean and standard deviation of the sampling distribution, you just need to plug in the values of μ, σ, and n into these formulas.

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How does one calculate mean and standard deviation of the sampling distribution.

How to interpret the standard deviation from part (1).

the wire has uniform density, the center of gravity is located at the centroid of the quarter-circle, which is (4/3, 4/3).

The center of gravity (x¯,y¯) of the wire lies on the line of symmetry, which passes through the origin and the centroid of the quarter-circle.

The centroid of a quarter-circle with radius 6 is located at (4/3, 4/3) from the origin (as derived using calculus). Thus, the line of symmetry passes through the origin and (4/3, 4/3).

The equation of the line passing through two points (x1, y1) and (x2, y2) is given by:

(y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)

Substituting (x1, y1) = (0, 0) and (x2, y2) = (4/3, 4/3), we get:

(y - 0) / (x - 0) = (4/3 - 0) / (4/3 - 0)

Simplifying, we get:

y = x

Therefore, the center of gravity (x¯,y¯) is located on the line y = x.

Since the wire has uniform density, the center of gravity is located at the centroid of the quarter-circle, which is (4/3, 4/3).

Hence, x¯=y¯=4/3.

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The equation that represent the amount of caffeine C in a pregnant woman’s body t hours  is [tex]C = 260(0.05)^{(t)}[/tex], under the condition that pregnant women process caffeine at about 5. 6% per hour.

Now the amount of caffeine C in a pregnant woman’s body t hours after the coffee is  ingested can be projected  by the equation

[tex]C = 260(0.05)^{(t)}[/tex]
here
C = amount of caffeine in milligrams (mg)
t = time in hours since the coffee was consumed
0.05 is the convention of 5.6%

It's crucial to note that pregnant women metabolize caffeine gradually slower than non-pregnant women, and it can take 1.5–3.5 times longer to eliminate caffeine from their body. In fact, most experts agree that caffeine is safe during pregnancy if limited to 200 mg or less per day.

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If the point starts at the π/2 position, the equation for the blade of the wind turbine will be sin(θ + π/2) = sin(θ - π/2)cos(β) + cos(θ - π/2)sin(β).

The equation for the blade of the wind turbine is given by the expression sin(θ) = sin(θ - β)cos(α) + cos(θ - β)sin(α), where θ represents the angle of the blade, β represents the angle between the wind direction and the blade, and α represents the pitch angle of the blade.

If the point starts at the π/2 position, we need to substitute θ + π/2 for θ in the equation. This gives us sin(θ + π/2) = sin(θ - β + π/2)cos(α) + cos(θ - β + π/2)sin(α).

Using trigonometric identities, we can simplify this expression to sin(θ + π/2) = cos(θ - β)cos(α) - sin(θ - β)sin(α).

Finally, substituting β for (π/2 - β) in the above equation, we get sin(θ + π/2) = sin(θ - π/2)cos(β) + cos(θ - π/2)sin(β). This is the required equation for the blade of the wind turbine if the point starts at the π/2 position.

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Answer:

Here we try the method of trial and error to find out if the equations have a common answer as zero

Step-by-step explanation:

Now,

(a) x²-4=0

(b) x²=-4

(c) 3x²+12=0

(d) 4x²=16

(e) 2(x-2)2=0

Here if we check the first equation i.e x²-4=0

Equating - 2²-4=4-4

=0

2²-4= 4-4

=0

So option (a) is true

Now x²=-4

Substituting, - 2²=4 & 2²=4

So here we get 4≠-4

Therefore, (b) is not true

Now 3x²+12=0

3(-2)²+12= 3(4)+12

=12+12= 24

3(2)²+12= 12+12

= 24

4x²=16

Substituting, 4(-2)²= 4(4)= 16

4(2)²= 4(4)= 16

So option (d) is also true

Now, 2(x-2)2=0

Substituting, 2(-2-2)2= 2(-4)2

4(-4)= - 16

2(2-2)2= 2(0)2

=4(0)=0

Here when we put x=-2, we get - 16

when we put x=2, we get 0

So the following equation is true only for x=2 and not x=-2

I hope this helps ;)

The particular solution Yp(t) is found by following these steps and combining it with the complementary solution Yc(t) to get the general solution Y(t) = Yc(t) + Yp(t).

To find the particular solution to [tex]y'' + y' - 12y = -340sin(t)[/tex] using variation of parameters, follow these steps:
1. Find the complementary solution Yc(t) for the homogeneous equation [tex]y'' + y' - 12y = 0[/tex]. The characteristic equation is[tex]r^2 + r - 12 = 0[/tex], which has roots r1 = -4 and r2 = 3. Thus, [tex]Yc(t) = C1e^(-4t) + C2e^(3t)[/tex].

2. Assume the particular solution Yp(t) is in the form [tex]Yp(t) = u1(t)e^(-4t) + u2(t)e^(3t)[/tex]. Calculate the Wronskian [tex]W(Y1, Y2) using Y1 = e^(-4t) and Y2 = e^(3t).[/tex]  3. Find u1'(t) and u2'(t) using the given nonhomogeneous equation: -340sin(t) = -Y1(t)u1'(t) + Y2(t)u2'(t).

4. Integrate u1'(t) and u2'(t) to find u1(t) and u2(t).
5. Plug u1(t) and u2(t) back into the particular solution [tex]Yp(t) = u1(t)e^(-4t) + u2(t)e^(3t[/tex]).

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The percent error is 4.76%.

To find the percent error, we first need to calculate the actual error, which is the absolute difference between the estimated points and the actual points:

Actual error = |252 - 240| = 12

Next, we need to calculate the percent error, which is the ratio of the actual error to the estimated value, expressed as a percentage:

Percent error = (actual error / estimated value) x 100%

Percent error = (12 / 252) x 100%

Percent error = 4.76%

Therefore, the percent error is 4.76%.

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The perimeter of the bedroom is 116ft

A rectangle is a type of quadrilateral, whose opposite sides are equal and parallel.

The perimeter of a rectangle is expressed as;

P = 2(l+w)

where l is the length and w is the width of the rectangle.

length = √ 21-21)²+ 18-(-7)²

= √25²

l = 25 ft

width = √ 21-(-12)²+18-18)²

= √ 33²

= 33

Perimeter = 2(33+25)

= 2 × 58

= 116ft

therefore the perimeter of the bedroom is 116ft.

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Answer:

1 candy bar is $1.25

2 candy bars is $2.50

3 candy bars is $3.75

4 candy bars is $5

Equation:

y = 1.25x

y being the cost and x being number of candy bars

The number of years apart between Eratosthenes was born in 276 BC (Before Christ) Abraham Lincoln died in 1865 is 2140 years.

Eratosthenes was born in 276 BC

Abraham Lincoln died in 1865 AD

BC = Before Christ

AD = anno dominie

The difference between the born date of Eratosthenes and the death date of Abraham Lincoln is to be calculated by

Year difference = BC year + AD year - 1

Year difference = 276 + 1865 - 1

Year difference = 2140

hence the  difference between the born date of Eratosthenes and the death date of Abraham Lincoln is 2140

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The total length of the string in feet is 6 feet, and the combined capacity of both water jugs in cups is 75 cups.

Step 1: To find the total length of the string in inches, Mara needs to add the lengths of the three pieces of string:

30 inches + 22 inches + 20 inches = 72 inches

So the total length of the string in inches is 72 inches.

Step 2: To convert inches to feet, we need to divide the number of inches by 12 (since there are 12 inches in a foot):

72 inches ÷ 12 = 6 feet

Therefore, once Mara puts together the three pieces of string, the total length will be 6 feet.

Step 1: To find out how many ounces of water both water jugs hold altogether, we need to add the capacity of the two jugs:

300 ounces + 300 ounces = 600 ounces

So both water jugs together can hold 600 ounces of water.

Step 2: To convert ounces to cups, we need to divide the number of ounces by 8 (since there are 8 ounces in a cup):

600 ounces ÷ 8 = 75 cups

Therefore, both water jugs together can hold 75 cups of water.

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A. Melissa's monthly payment is approximately $146.89.

B. The total amount of the monthly payments over the four-year period is approximately $7,052.72.

C. The finance charge on the loan is approximately $1,052.72.

The interest on a loan or principal amount can be easily calculated using simple interest. Simple interest is a notion that is employed across a wide range of industries, including banking, finance, automobiles, and more.

To calculate the monthly payment, we can use the formula:

Monthly payment = [P x r x (1 + r)ⁿ] / [(1 + r)ⁿ⁻¹]

where P is the principal (the amount of the loan), r is the monthly interest rate, and n is the number of monthly payments.

A. To find the monthly payment, we first need to calculate the values of r and n.

Since the APR is 10%, the monthly interest rate is 10% / 12 = 0.00833 (rounded to 5 decimal places).

The number of monthly payments over four years is 4 x 12 = 48.

Using these values, we can calculate the monthly payment:

Monthly payment = [6000 x 0.00833 x (1 + 0.00833)⁴⁸] / [(1 + 0.00833)⁴⁸⁻¹]

Monthly payment ≈ $146.89

So Melissa's monthly payment is approximately $146.89.

B. To find the total amount of the monthly payments, we can multiply the monthly payment by the number of payments:

Total amount of monthly payments = Monthly payment x Number of payments

Total amount of monthly payments = $146.89 x 48

Total amount of monthly payments ≈ $7,052.72

So the total amount of the monthly payments over the four-year period is approximately $7,052.72.

C. To find the finance charge, we can subtract the original principal from the total amount of the monthly payments:

Finance charge = Total amount of monthly payments - Principal

Finance charge = $7,052.72 - $6,000

Finance charge ≈ $1,052.72

So the finance charge on the loan is approximately $1,052.72.

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The depth of trail mix box that is shaped like a rectangular prism must be 2.4 inches.

We are going to use the formula for determining volume to work out the depth of the rectangular prism.

The volume of a rectangular prism is given by the formula:

V = l × w × h

where:

V = the volume

l = the length

w = the width

h = the height.

Given:

w = 7.5 inches

h = 9 inches

V = 162 cubic inches

Now, we are to find the value of d, the depth of the box, which corresponds to the length of the rectangular prism.

Substituting the values into the formula for volume:

162 = d × 7.5 × 9

Simplifying the right-hand side:

162 = 67.5d

Dividing both sides by 67.5, we get:

d = 162 ÷ 67.5

d = 2.4 inches

Therefore, the depth of the box shaped as rectangular prism = 2.4 inches.

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Reactions at points A, C, and F are as follows: RA = -RC, RAy = RCy, and RF = 10 kN.

To determine reactions at points A, C, and F for the assembly consisting of two 10mm diameter rods (A-B and C-D), a 20mm diameter rod (E-F), and a rigid bar (H), with point G located midway between points E and F, and a force of 10 kN applied:

First, calculate the cross-sectional areas of the rods:
A1 = π(10mm)² / 4 = 78.54 mm² (for rods A-B and C-D)
A2 = π(20mm)² / 4 = 314.16 mm² (for rod E-F)

Next, calculate the effective modulus of elasticity for each rod:
E1 = E2 = 101 GPa

Now, use equilibrium equations to determine reactions at points A, C, and F:
ΣFx = 0: RAx + RCx = 0
ΣFy = 0: RAy + RCy + RF = -10 kN
ΣMG = 0: (RAy)(L/2) - (RCy)(L/2) = 0

Solving the equilibrium equations, we get:
RAx = -RCx
RAy = RCy
RF = 10 kN

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