A 10​-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 6 ft from the​ house, the base is moving away at the rate of 24 ​ft/sec.
a. What is the rate of change of the height of the top of the​ ladder?
b. At what rate is the area of the triangle formed by the​ ladder, wall, and ground changing​ then?
c. At what rate is the angle between the ladder and the ground changing​ then?

The y-intercept of the graph is: C. "the temperature in degrees Fahrenheit when the temperature is zero degrees Celsius."

In mathematical terms, the y-intercept is the point where a line or curve intersects the y-axis on a coordinate plane.

In practical terms, the y-intercept of a situation often represents the starting point or initial value of a variable or function.

In the graph given above that shows the relationship between temperature in degrees Fahrenheit and degrees Celsius, the y-intercept is 32 degrees Fahrenheit, which means: C. "the temperature in degrees Fahrenheit when the temperature is zero degrees Celsius."

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The answer is option C) 30,700

The problem involves using linear interpolation to estimate the population of Little Elm in 2020 based on the given data from 2010 and 2015.

To start, we need to calculate the annual growth rate of the population between 2010 and 2015. This can be done by taking the difference between the population in 2015 and the population in 2010, and then dividing that difference by the number of years in that time period:

Annual growth rate = (17,150 - 3,646) / 5 = 2,700

This means that, on average, the population of Little Elm grew by 2,700 people per year between 2010 and 2015.

Next, we can use this annual growth rate to estimate the population of Little Elm in 2020. To do this, we need to add 10 years of growth to the initial population of 3,646 (since we want to estimate the population in 2020, which is 10 years after 2010):

Estimated population in 2020 = 3,646 + (2,700 x 10) = 30,646

Therefore, the estimated population of Little Elm in 2020 is approximately 30,700, which corresponds to answer choice C.

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Inflation is an important factor to consider when planning future purchases.

The Inflation Calculator does not provide the inflation-adjusted price for a fancy leather couch 10 years from now because it relies on historical data and cannot predict future inflation rates accurately.

Inflation rates are influenced by various economic factors and can fluctuate over time.

Therefore, it's difficult to determine the exact future cost of items with certainty. However, you can use historical trends as a general guide to estimate potential increases in cost due to inflation.

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The solution to the equation is x = 15.

To rewrite the exponential equation using the same base, we need to express both 8 and 32 as powers of the same base. Since both 8 and 32 are powers of 2, we can rewrite the equation as:

[tex](2^3)^(2x) = (2^5)^(x+3)[/tex]

Here, we used the fact that[tex](a^b)^c = a^(b*c)[/tex]to simplify the exponents. We also used the property that 8 is equal to 2 raised to the power of 3, and 32 is equal to 2 raised to the power of 5.

Now that we have rewritten the equation with the same base, we can equate the exponents on both sides of the equation to solve for x:

[tex]2^(6x) = 2^(5x + 15)[/tex]

Since the bases on both sides of the equation are equal, we can equate the exponents and solve for x:

6x = 5x + 15

Subtracting 5x from both sides, we get:

x = 15

Therefore, the solution to the equation is x = 15.

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The two digit Combinations that never appear in any of the passcodes are 5 and 6.

There are 49 different passcodes that can be used. The sum of the six digits of the current passcode is 81, which is a square number. Therefore, the current passcode is made up of three different two-digit square numbers with the sum of their digits equal to 9.

The possible combinations are

(16, 25, 40),

(16, 49, 16),

(25, 36, 20),

(36, 49, 4),

(49, 64, 9),

(64, 81, 16), and

(81, 16, 64).

The only combination that has the same first and sixth digits is (16, 25, 40), so the current passcode is 162540.

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Answer:

Step-by-step explanation:

60 x 3 = 180

180 miles

The measure of the base angle x is approximately 81.54 degrees.

To get the measure of the base angle, we can use the fact that the sum of the angles in a triangle is always 180 degrees. Since this is an isosceles triangle, we know that the two base angles are congruent (they have the same measure).
Let's call the measure of each base angle y. Then we can set up an equation:
y + y + x = 180
Simplifying, we get:
2y + x = 180
Now we can use the fact that the legs of the triangle are congruent to find the measure of y. Since this is an isosceles triangle, we know that the two legs are congruent. This means we can use the Pythagorean theorem to find the length of the height, h, of the triangle:  h^2 = 9^2 - (12/2)^2
h^2 = 81 - 36
h^2 = 45
h = sqrt(45)
h = 6.71 (rounded to two decimal places)
Now we can use the definition of the tangent function to find y:
tan(y) = h / (12/2)
tan(y) = 6.71 / 6
tan(y) = 1.1183                                                                                                                                                                                     y = tan^-1(1.1183)
y = 49.23 degrees (rounded to two decimal places)
Finally, we can substitute this value of y into our equation to find x:
2y + x = 180
2(49.23) + x = 180
98.46 + x = 180
x = 81.54 degrees (rounded to two decimal places)
Therefore, the measure of the base angle x is approximately 81.54 degrees.

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The slope of the secant line connecting two points on a curve can be calculated using the formula: slope of secant line = (change in y) / (change in x).

For this problem, the two points are (-1, f(-1)) and (2, f(2)), and we need to find the slope of the secant line connecting them. To do this, we first need to find the y-coordinates of these points by plugging the given x-values into the function f(x) = 3x^2 + 4x - 5. Then we can use the formula for the slope of a secant line to find the slope of the line connecting these two points. The slope of the secant line is an important concept in calculus and is used to approximate the instantaneous rate of change of a function at a particular point.

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Answer:

14

Step-by-step explanation:

Cam put a total of 4 12/25 pounds of rice in bags, which is equivalent to 112/25 pounds of rice.

Each bag weighs 7/25 pound, so the total number of bags of rice is:

(112/25) ÷ (7/25) = 16

Cam uses 1/8 of the bags of rice, which is:

(1/8) x 16 = 2

So Cam uses 2 bags of rice.

The number of bags of rice left is the total number of bags (16) minus the number of bags used (2):

16 - 2 = 14

Therefore, 14 bags of rice are left.

Answer:

Step-by-step explanation:

5^2 + x^2 = 8^2

25 + x^2 = 64

x = 6.2

The domain of the function f(x) = 1/x - 8 is (-∞, 0) U (0, +∞), and f = 1/x - 8.

The function f(x) is defined as 1/x - 8. The domain of the function is the set of all possible values of x for which the function is defined. Since the function involves division by x, x cannot be equal to zero. Therefore, the domain of f(x) is (-∞, 0) U (0, +∞), which means that x can take any value except 0.

To find f(x), we simply substitute the expression for f(x) in the definition of the function. Therefore, we have:

f(x) = 1/x - 8

This is the final answer. We cannot simplify it any further. The function f(x) represents a hyperbola with a vertical asymptote at x = 0 and a horizontal asymptote at y = -8.

As x approaches 0 from the left, f(x) goes to negative infinity, and as x approaches 0 from the right, f(x) goes to positive infinity. Similarly, as x approaches positive or negative infinity, f(x) approaches 0.

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The probability is that out of 12 customers, 7 will order a non-alcoholic beverage, and the remaining 5 will order an alcoholic beverage.

The likelihood that a client will arrange a non-alcoholic refreshment is given as 48%, which implies that the likelihood that a client will arrange an alcoholic refreshment is (100 - 48) = 52%.

Out of 12 customers, 5 will arrange liquor, which suggests that the remaining clients will arrange a non-alcoholic refreshment. We are able to calculate the number of clients who will arrange a non-alcoholic refreshment as takes after:

Number of clients who will arrange a non-alcoholic refreshment =

Add up to a number of clients - Number of clients who will arrange liquor

= 12 - 5

= 7

Subsequently, out of 12 clients, 7 will arrange a non-alcoholic refreshment, and the remaining 5 will arrange an alcoholic refreshment.

It is critical to note that these calculations are based on the presumption that each client will as it were arrange one refreshment. 

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Answer: C

Step-by-step explanation:

The weight of the goods on the ship decreased by 50 percent.

To find the percentage decrease in the weight of goods on the ship, we need to calculate the difference between the initial weight and the final weight, divide it by the initial weight, and then multiply by 100 to get the percentage decrease.

The initial weight of the goods on the ship was 60,010 tons and the final weight after unloading was 30,005 tons.

The difference between the initial weight and the final weight is 60,010 - 30,005 = 30,005 tons.

To find the percentage decrease, we divide the difference by the initial weight:

30,005 / 60,010 = 0.5

Multiplying by 100 gives us the percentage:

0.5 x 100 = 50%

Therefore, the weight of goods on the ship has decreased by 50 percent.

In conclusion, the percentage decrease in the weight of goods on the ship is 50 percent. This means that the ship has unloaded half of its initial weight and now carries only half of the weight it carried when it left the port last week. This calculation can be helpful for the shipping company to determine the efficiency of their transportation and to plan for future shipments.

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The measurement of BC in the given figure is 4√2.

Given is figure we need to find the measurement of BC,

∠GBC = ∠CDF = 45° [alternate angles]

So, in triangle DCF,

Cos 45° = DF/DC

1/√2 = DF/12√2

DF = 12

Now we see that the triangles DCF and BCG are similar triangles by AA rule,

So, according to the definition of similar triangles,

DC/BC = DF/BG

12√2/BC = 12/(7-3)

12√2/BC = 12/4

12√2/BC = 3

3BC = 12√2

BC = 4√2

Alternatively, you can find the value of BC, using the trigonometric ratios in triangle GBC,

Cos 45° = GB/BC

GB = 7-3 = 4

Therefore,

Cos 45° = 4/BC

1/√2 = 4/BC

BC = 4√2

Hence the measurement of BC in the given figure is 4√2.

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The x-coordinate of the chicken coop is 1, which means that the chicken coop is located at the point (1, 0).

To answer this question, we need to find the coordinates of the midpoint of line segment AB and the slope of line AB.

a) To find the midpoint of line segment AB, we use the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.

Substituting the values, we get:

Midpoint = ((-2 + 4)/2, (4 - 4)/2)

Midpoint = (1, 0)

Therefore, the midpoint of line segment AB is (1, 0).

b) To find the slope of line AB, we use the slope formula:

Slope = (y2 - y1)/(x2 - x1)

Substituting the values, we get:

Slope = (-4 - 4)/(4 - (-2))

Slope = (-8)/(6)

Slope = -4/3

Therefore, the slope of line AB is -4/3.

Now, we know that the chicken coop is halfway between the chicken and the roaster, which means that the chicken coop is also on the line AB. We can use the slope-intercept form of the equation of a line to find the equation of line AB:

y = mx + b

where m is the slope of the line, and b is the y-intercept.

Substituting the values of slope and midpoint, we get:

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

Therefore, the equation of line AB is y = (-4/3)x + (4/3).

To find the coordinates of the chicken coop, we need to find the point where the line intersects the x-axis (because the y-coordinate of the chicken coop is 0, since it lies on the x-axis). To do this, we set y = 0 in the equation of line AB:

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

4/3 = (4/3)x

x = 1

Therefore, the x-coordinate of the chicken coop is 1, which means that the chicken coop is located at the point (1, 0).

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The distance between the chicken  at A(-2,4) and the roasted is at B(4,-4).

To determine the chicken of the line that contains points A(-2,4) and B(4,-4), we can use the slope formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the coordinates, we get:

slope = (-4 - 4) / (4 - (-2)) = -8 / 6 = -4 / 3

So the slope of the line is -4/3.

To find the equation of the line, we can use the point-slope form:

y - y1 = m(x - x1)

Substituting one of the points and the slope, we get:

y - 4 = (-4/3)(x - (-2))

Simplifying, we get:

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

Therefore, the equation of the line that contains points A(-2,4) and B(4,-4) is y = (-4/3)x + 4/3.

To find the distance between the chicken and the roaster, we can use the distance formula:

d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]

Substituting the coordinates, we get:

d = sqrt[(4 - (-2))^2 + (-4 - 4)^2] = sqrt[6^2 + (-8)^2] = sqrt[100] = 10

Therefore, the distance between the chicken and the roaster is 10 units.

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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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If a population is normal with a variance of 99, the necessary sample size is 7 (Option B).

To find the necessary sample size for a given confidence level and margin of error, we can use the formula:

n = (Z² * σ²) / E²

where n is the sample size, Z is the Z-score corresponding to the desired confidence level, σ² is the population variance, and E is the margin of error.

In this case, the population variance (σ²) is 99, the desired confidence level is 68.26%, and the margin of error (E) is 4 units. The Z-score corresponding to a 68.26% confidence level is approximately 1, as it is close to one standard deviation from the mean in a normal distribution.

Now, we can plug the values into the formula:

n = (1² * 99) / 4²
n = (1 * 99) / 16
n = 99 / 16
n ≈ 6.19

Since we cannot have a fraction of a sample, we round up to the nearest whole number, which is 7. So, the necessary sample size is 7 (Option B).

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c. rotate it 90° counterclockwise about point (-3,-3), then translate it 8 units to the right

The correct pair of transformations that moves quadrilateral 1 to quadrilateral 2 is option d: translate it 8 units to the right, then reflect it over the line y = -3.

First, by translating it 8 units to the right, all points of quadrilateral 1 will shift horizontally to the right by 8 units, maintaining their relative positions.

Next, reflecting it over the line y = -3 will result in a vertical flip of the shape. This reflection will change the orientation of the quadrilateral while keeping the translated positions intact.

Together, these two transformations will precisely move quadrilateral 1 to quadrilateral 2. Therefore, option d is the correct choice.

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Answer:

Step-by-step explanation:

6.48x1.0e5=x

x/0.35

Answer:

Problem 8:
Option C)       DE = √21

Problem 9:
Option C)     Perimeter = 30 cm

Step-by-step explanation:

1. Problem 8

Draw a line segment connecting the center of the circle, X, to A. This is the radius of the circle

The points ABX form a right triangle with AX as the hypotenuse and AB, BX as the legs of the right triangle

By the Pythagorean theorem
hypotenuse² = sum of the squares of the two legs

Plugging in line segment references

AX² = AB² + BX²

We are given AC = 8, BX = 3

Since the segment BX intersects AC at right angles, AB = BC = AC/2

So AB = 8/2= 4

Plugging these values into the Pythagorean formula
AX² = AB² + BX²

AX² = 4² +3²

AX² = 16 + 9

AX² = 25

Draw a line connecting X and D
The points DEX form a right triangle with DX as the hypotenuse and EX and DE as the legs

Again, by the Pythagorean theorem

DX² = DE² + EX²

But DX is the radius of the circle so it must be the same as AX

Substitute for DX in terms of AX
DX² = DE² + EX²

=> AX² = DE² + EX²

But AX² = 25 and EX = 2 giving EX² = 4

Therefore

AX² = DE² + EX² becomes
25 = DE² + 4

DE² = 25 - 4

DE² = 21

DE = √21

This is choice C

Problem 9

To find the perimeter, just add up the individual side lengths:

Perimeter = 10 + 8 + 7 + 5 = 30 cm

Option C

Gray's $3,500 tax-deductible contribution to his IRA will save him $980 on his taxes.

When Gray contributes $3,500 to his individual retirement account (IRA), it is considered a tax-deductible contribution. This means that the amount contributed is deducted from his taxable income, reducing the amount of taxes he owes.

Since Gray is in the 28 percent tax bracket, this means that for every dollar of taxable income, he pays 28 cents in taxes. To calculate the tax savings from his $3,500 IRA contribution, we need to multiply the contribution amount by his tax rate:

$3,500 (contribution) x 0.28 (tax rate) = $980 (tax savings)

In this case, Gray's $3,500 tax-deductible contribution to his IRA will save him $980 on his taxes. By contributing to his IRA, Gray not only invests in his future retirement but also takes advantage of the tax benefits associated with these accounts. In the end, he reduces his taxable income and, consequently, the amount of taxes he needs to pay.

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A function that forms an arithmetic sequence include the following: D. F(x) = 2x - 4.

In Mathematics and Geometry, the nth term of an arithmetic sequence can be calculated by using this equation:

aₙ =  a₁ + (n - 1)d

Where:

Next, we would determine the common difference as follows.

Common difference, d = a₂ - a₁

Common difference, d = -6 + 8 = -4 + 6 = -2 + 4

Common difference, d = -2.

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Answer:

In simplified form: 1/27

Evaluated: 0.037

Step-by-step explanation:

To simplify and evaluate 81^(-3/4), we use the rule that (a^m)^n = a^(mn) and rewrite the expression as (3^4)^(-3/4). Then, we use the rule that a^(-n) = 1/(a^n) to get:

81^(-3/4) = (3^4)^(-3/4) = 3^(-3) = 1/(3^3) = 1/27

Therefore, 81^(-3/4) simplifies to 1/27 and evaluates to 0.037

Answer:

Step-by-step explanation:

Step-by-step explanation:

d = a2 - a1

d = 11 - 5

d = 6

[tex]Sn \: = \frac{n}{2} (2a1 + (n - 1)d) \\ S7 = \frac{7}{2} (2 \times 11 + (7 - 1)6) \\ S7 = \frac{7}{2} (22 + 36) \\ S7 = \frac{7}{2} \times 58 \\ S7 = 7 \times 29 \\ S7 = 203[/tex]

The square root of the sample size (n=37), which is approximately 0.820 seconds:

X ~ N(8, 5/√37)

The sample distribution of the mean, denoted as X, is the distribution of all possible sample means of a certain sample size taken from a population. In this case, we have a population of couples in conflict, and we have a sample size of 37. The mean length of time couples in conflict listen to each other is known to be 8 seconds with a standard deviation of 5 seconds.

Since we have a large enough sample size (n=37), we can assume that the sample mean follows a normal distribution according to the Central Limit Theorem. The Central Limit Theorem states that the sample means from a large enough sample size will be approximately normally distributed regardless of the population distribution, given that the samples are independent and randomly selected.

Thus, the sample distribution of X can be represented by a normal distribution with a mean of 8 seconds (same as the population mean) and a standard deviation of 5 seconds divided by the square root of the sample size (n=37), which is approximately 0.820 seconds:

X ~ N(8, 5/√37)

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Answer:

Para comparar las propuestas de los bancos, debemos llevar las tasas de interés a una misma unidad de tiempo. Podemos convertir la tasa del Banco A de bimestral a mensual multiplicándola por 2 (ya que hay 6 bimestres en 1 año):

Tasa del Banco A: 1,5% * 2 = 3% mensual

Tasa del Banco B: 0,5% mensual

Para calcular los intereses que se obtendrán en cada banco, podemos utilizar la fórmula del interés compuesto:

I = C * ((1 + r/n)^(n*t) - 1)

Donde:

I es el interés

C es el capital inicial

r es la tasa de interés en forma decimal

n es el número de veces que se capitaliza al año

t es el tiempo en años

Para el Banco A, como la tasa está en meses, capitalizaremos mensualmente (n=12):

I = 40 000 * ((1 + 0,03/12)^(12*2) - 1) = 4 896,18 soles

Para el Banco B, como la tasa ya está en meses, capitalizaremos mensualmente (n=12):

I = 40 000 * ((1 + 0,005)^(12*2) - 1) = 4 225,48 soles

Por lo tanto, la propuesta más conveniente es la del Banco A, ya que ofrece una tasa de interés mayor y genera un interés total de 4 896,18 soles. En cambio, el Banco B genera un interés total de 4 225,48 soles.

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D (5,2) and (2,5)

This is correct indeed