In an arithmetic sequence, the sum of the first 8 terms is 88. The product of the 1st and the 8th terms is 120. Calculate values for the 1st term.​

An equation for a rational function with the given characteristics is found as:  f(X) = a(x+4)(x) / (x+2)(x-4).

As x = -2 and X = 4 have vertical asymptotes, the denominator will contain the equations (x+2) and (x-4)

The terms (x+4) and 1 will be in the numerator since the x intercepts are (-4,0) , respectively (x-0).

Our current formula is f(X) = a(x+4)(x) / (x+2)(x-4)

We examine the horizontal asymptote to determine the value of A.

The limit of a function as x -> infinity would be - 3 since the horizontal asymptote reflects what the function looks like as x approaches infinity.

Thus,  an equation for a rational function with the given characteristics is found as:  f(X) = a(x+4)(x) / (x+2)(x-4)

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Answer:

w = 6in, l = 10in

Step-by-step explanation:

Area of a rectangle is [tex]A=l*w[/tex]

and [tex]l = 4 + w[/tex]

We can solve for length and width by substituting these values for the equation.

[tex]60in^2=(4+w)*w[/tex]

[tex]60in^2=4w+w^2[/tex]

[tex]0 = w^2 + 4w + 60in^2[/tex]

factor out the equation.

[tex]0= (w+10)(w-6)[/tex]

therefore w = 6in (tossing out the negative solution)

which means

[tex]l = 4 + w[/tex]

[tex]l = 6in + 4[/tex]

l = [tex]10in[/tex]

Hope this helps!
Brainliest is much appreciated!

Let x be the number of text messages that your older sister sends per day. Then we can write an equation based on the given information:

x + 12 = 26

We can solve for x by subtracting 12 from both sides of the equation:

x = 26 - 12

x = 14

Therefore, your older sister sends about 14 text messages per day.

Answer:

Rounding to the nearest hundredth, the area of the shaded region is approximately 167.91 square yards.

Step-by-step explanation:

To find the area of the shaded region between two circles with the same center, we need to subtract the area of the smaller circle from the area of the larger circle.

The area of the larger circle is:

A_outer = πr_outer^2

= π(11.9 yd)^2

≈ 445.03 yd^2

The area of the smaller circle is:

A_inner = πr_inner^2

= π(9.4 yd)^2

≈ 277.12 yd^2

Therefore, the area of the shaded region is:

A_shaded = A_outer - A_inner

≈ 445.03 yd^2 - 277.12 yd^2

≈ 167.91 yd^2

Hope this helps! Sorry if it doesn't. If you need more help, ask me! :]

Answer: x=7 y=5

Step-by-step explanation:

I believe this is the answer to the question  

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{annually}, thus once} \end{array}\dotfill &1\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.04}{1}\right)^{1\cdot 6} \implies A \approx 3795.96 \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{quarterly}, thus four} \end{array}\dotfill &4\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.04}{4}\right)^{4\cdot 6} \implies A \approx 3809.20 \\\\[-0.35em] ~\dotfill[/tex][tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{monthly}, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000\left(1+\frac{0.04}{12}\right)^{12\cdot 6} \implies A \approx 3812.23 \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$3000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ t=years\dotfill &6 \end{cases} \\\\\\ A = 3000e^{0.04\cdot 6} \implies A \approx 3813.75[/tex]

The median shows that the average number of students per middle school classroom in both districts is 24, while the mean shows that the average number of students per middle school classroom in one district is 24 and the other is 25.  

Median is the middle value of a set of data. It is the value that divides the data set into two equal halves. It is an important measure of central tendency and is used to compare sets of data.  

The difference between the mean number of students per middle school classroom in each school district is 2 students. The difference between the median number of students per middle school classroom in each school district is 0 students.

Based on the number of students shown for each school district, the median would provide the most accurate picture of the number of students in a middle school classroom, as there are some extreme values that would affect the mean but not the median.

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Applying the definition of variation, we have:

1. When x = 36, y = 6.

2. When x = 1, y = 27.

3. It takes 3 seconds for the object to fall 144 feet.

4. Length of a string that vibrates 64 times per second is 48 inches.

Inverse variation is a type of relationship between two variables in which one variable increases while the other decreases, or vice versa, in a way that the product of the two variables remains constant.

In other words, if we have two variables x and y that are inversely proportional, we can write:

x × y = k, where k is a constant.

1. We are given that y varies directly as the square root of x. This means that we can write:

y = k√x

where k is a constant of proportionality. To find the value of k, we can use the given information that when x = 16, y = 4:

4 = k√16

4 = 4k

k = 1

Now we can use this value of k to find y when x = 36:

y = 1√36

y = 6

Therefore, when x = 36, y = 6.

2. We are given that y varies inversely with the cube of x. This means that we can write:

y = k/x³

where k is a constant of proportionality. To find the value of k, we can use the given information that when x = 3, y = 1:

1 = k/3³

1 = k/27

k = 27

Now we can use this value of k to find y when x = 1:

y = 27/1³

y = 27

Therefore, when x = 1, y = 27.

3. We are given that the distance that an object falls varies directly with the square of the time, t, of the fall. This means that we can write:

d = kt²

where d is the distance, k is a constant of proportionality, and t is the time. To find the value of k, we can use the given information that when t = 1, d = 16:

16 = k(1)²

16 = k

k = 16

Now we can use this value of k to find the time it takes for the object to fall 144 feet:

144 = 16t²

9 = t²

t = 3

Therefore, it takes 3 seconds for the object to fall 144 feet.

4. We are given that the rate of vibration of a string under constant tension varies inversely with the length of the string. This means that we can write:

r = k/L

where r is the rate of vibration, L is the length of the string, and k is a constant of proportionality.

To find the value of k, we can use the given information that when L = 24 inches, r = 128 vibrations per second:

128 = k/24

k = 3072

Now we can use this value of k to find the length of a string that vibrates 64 times per second:

64 = 3072/L

L = 48 inches

Therefore, the length of a string that vibrates 64 times per second is 48 inches.

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Answer:

y= -x - 3 is correct

Step-by-step explanation:

So, Jane's hourly rate is $6.00 per hour.

In mathematics, division is a basic arithmetic operation that involves splitting a number or quantity into equal parts or groups. It is the inverse operation of multiplication, where we find out how many times one number (the divisor) can fit into another number (the dividend) without leaving any remainder.

Here,

However, we can calculate her hourly rate by dividing her total earnings by the number of hours she worked.

From the payroll information provided, Jane earned $120.00 for 20 hours of work. Therefore, her hourly rate would be:

Hourly rate = Total earnings / Number of hours worked

Hourly rate = $120.00 / 20 hours

Hourly rate = $6.00 per hour

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Answer:

a

Step-by-step explanation:

The formula to calculate simple interest is:

I = P * r * t

Where:

I = Interest

P = Principal amount

r = Rate of interest

t = Time period

Given:

Principal amount (P) = $1,700

Rate of interest (r) = 8% per annum

Time period (t) = 4 years

Using the formula of simple interest:

I = P * r * t

I = 1,700 * 0.08 * 4

I = $544

Therefore, the total interest on a 4-year term loan of $1,700 with a simple annual interest rate of 8% is $544.

Answer: A. $544

To adjust the recipe to make 30 rolls instead of 12, we need to multiply the amount of each ingredient by a factor of 30/12 = 2.5.

A factor is a number or expression that divides another number or expression evenly without a remainder.

To adjust the recipe to make 30 rolls instead of 12, we need to multiply the amount of each ingredient by a factor of 30/12 = 2.5.

To determine this factor, we divide the desired number of rolls (30) by the original number of rolls (12):

2.5 = 30/12

To find the amounts of each ingredient needed to make 30 rolls, we multiply the original amounts by 2.5:

Yeast: 1/4 oz x 2.5 = 0.625 oz (or about 0.63 oz)

Warm water: 1 cup x 2.5 = 2.5 cups

Molasses: 3 tbsp x 2.5 = 7.5 tbsp (or about 1/2 cup + 1 tbsp)

Powdered milk: 1/3 cup x 2.5 = 0.83 cups (or about 3/4 cup + 1 tbsp)

Egg yolk: 1 beaten yolk x 2.5 = 2.5 beaten yolks (or about 2 yolks)

Egg white: 1 beaten white x 2.5 = 2.5 beaten whites (or about 2 whites)

Bread flour: 3 1/2 cups x 2.5 = 8.75 cups (or about 8 3/4 cups)

Butter: 6 tbsp x 2.5 = 15 tbsp (or about 1 cup - 1 tbsp)

Salt: 1 tsp x 2.5 = 2.5 tsp (or about 2 1/2 tsp)

Whole wheat flour: 1/2 cup x 2.5 = 1.25 cups (or about 1 1/4 cups)

Raisins: 1/3 cup x 2.5 = 0.83 cups (or about 3/4 cup + 1 tbsp)

To make 30 rolls, you will need 0.625 oz (or about 0.63 oz) of yeast.

To make 30 rolls, you will need 0.83 cups (or about 3/4 cup + 1 tbsp) of powdered milk.

To make 30 rolls, you will need 15 tbsp (or about 1 cup - 1 tbsp) of butter.

To make 30 rolls, you will need 8.75 cups (or about 8 3/4 cups) of bread flour, which is 8.75 - 1.25 = 7.5 cups more than the 1.25 cups of whole wheat flour needed.

If you put in 3/8 cup of water instead of the 1/4 cup called for in the original recipe, you have added an extra 3/8 - 1/4 = 1/8 cup of water, or 2 tablespoons of water.

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Using set theory, the element in each of the set are, A - B = {3, 5}, (A - B)^c = {2, 4, 6, 7, 8, 9, 10, 11}, U - (A - B)^c = {3, 5}, (A \cup B)^c = {2, 4, 6} and (A^c U B^c)^c = {7, 9, 11}.

The given sets are:

U = {2,3,4,5,6,7,8,9,10,11}

A = {3,5,7,9,11}

B = {7,8,9,10,11}

We can use these sets to find the following:

1. A - B: The elements in set A that are not in set B are {3, 5}.

2. (A - B)^c: The complement of set A - B in U contains all the elements in U that are not in A - B. These elements are {2, 4, 6, 7, 8, 9, 10, 11}.

3. U - (A - B)^c: This set contains all the elements in U that are not in the complement of A - B. These elements are {3, 5}.

4. (A U B)^c: This set contains all the elements in U that are not in the union of sets A and B. The union of A and B is {3, 5, 7, 8, 9, 10, 11}, so the complement is {2, 4, 6}.

5. (A^c U B^c)^c: This set contains all the elements in U that are in neither the complement of set A nor the complement of set B. The complement of set A is {2, 4, 6, 8, 10} and the complement of set B is {2, 3, 4, 5, 6}. The union of these two sets is {2, 3, 4, 5, 6, 8, 10}, so the complement of this set in U is {7, 9, 11}.

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We can rewrite 3.1672 × 10¹² as 3.1672 × 10² The correct option is A

First we first express it as a number between 1 and 10 multiplied by a power of 10. In this case, 316.72 can be expressed as:

3.1672 × 10¹² (since we move the decimal point 2 places to the right to get to 31672)

However, in scientific notation, the exponent should represent the number of places the decimal point is moved. Since we moved the decimal point to the right, the exponent should be negative.

Therefore, we can rewrite 3.1672 × 10¹² as:

3.1672 × 10²

So, the answer is (A).

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Tt would take Maurice and Olinda approximately 47.37 minutes

What is work done?

Work done in physics is the dot product of force and displacement. When a force applied on an object results in a displacement, it is said to be work is done on the object.

Let's denote the time it takes both Maurice and Olinda working together to weed the garden as t.

We can use the formula: 1 / t = 1 / a + 1 / b

Plugging in the given values, we get: 1 / t = 1 / 45 + 1 / 50

1 / t = (50 + 45) / (50 * 45) = 95 / 2250

Then, we can invert both sides of the equation to get: t / 1 = 2250 / 95

Simplifying the right-hand side by dividing both the numerator and the denominator by 5, we get:

t = 2250 / 95 = 47.37 minutes (rounded to two decimal places)

Therefore, it would take Maurice and Olinda approximately 47.37 minutes to weed the garden if they work together.

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Answer:

Step-by-step explanation:

[tex]S=40R+20[/tex]

Answer:

Step-by-step explanation:

[tex]V=\pi r^2 h[/tex]

Substitute [tex]h=l,r=2,V=150,[/tex] we get

[tex]150=\pi \times 2^2 \times l[/tex]

[tex]150=4\pi l[/tex]

[tex]l=\frac{150}{4\pi}=11.94cm[/tex]

The point in which f(x) have a horizontal tangent are (-7 + √46)/3, (-7 - √46)/3

To find the values of x where the graph of f(x) has a horizontal tangent, we need to find where the derivative of f(x) is equal to zero (since the derivative gives the slope of the tangent line).

First, we find the derivative of f(x):

f'(x) = 3x² + 14x + 1

Next, we set f'(x) equal to zero and solve for x:

3x² + 14x + 1 = 0

Using the quadratic formula, we get:

x = (-14 ± √(14² - 431)) / (2*3)

x = (-14 ± √(184)) / 6

x = (-14 ± 2√46) / 6

Simplifying:

x = (-7 ± √46) / 3

Therefore, the values of x where the graph of f(x) has a horizontal tangent are (-7 + √46)/3 and (-7 - √46)/3.

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The correct statement regarding the relative minimums of each function is given as follows:

The minimum of h(x) is farther left and down from the minimums of f(x) and g(x).

The function f(x) is defined as follows:

f(x)= √x.

It has a relative minimum at (0,0), which is one the first quadrant.

The function g(x) is defined as follows:

√x-3+1

It is a translation of 3 units right and 1 unit up of f(x), hence the minimum has coordinates (3,1), which is on the first quadrant.

The function h(x) is defined as follows:

h(x)= √x+1-2

It is a translation of 1 unit left and 2 units down of f(x), hence the minimum has coordinates (-1, -2), which is on the third quadrant.

The third quadrant is farther left and down from the first quadrant.

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The time between the injections that is injected again when the dosage reaches 9 is solved to be 1.8 to the nearest tenth

The information given by the problem include the expression

D(h) = 20e^(-0.45h)

The expression is an exponential function that shows the relationship between time and the dosage in milligrams  

given that D = 9 we find h

D(h) = 20e^(-0.45h)

9 = 20e^(-0.45h)

9/20 = e^(-0.45h)

take ㏑ of both sides

㏑(9/20) = -0.45h

isolating h by dividing by -0.45

h = ㏑(9/20) / -0.45

h = 1.774

h = 1.8 to the nearest tenth

This shows that the time is 1.8 hours

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Answer: -5 1/4

Step-by-step explanation:

After answering the final question incorrectly, Sandra's score will be -5 1/4 in fraction form, and -5.25 in decimal form. This is because every time Sandra gets a question wrong her grade goes down by -3/4 points. And -4 1/2 minus -3/4 is -5 1/4. I hope this helped. If not, let me know and I can try to reword it.

Answer: b

Step-by-step explanation:

Answer:

A equals 13.5 centimeters

To determine the distance of the foot of the ladder from the wall, we can use trigonometry. Let's call the distance we want to find "x".

First, let's draw a diagram:

          |\

          | \

   20   |  \ x

          |   \

          |____\

           40°

In this diagram, the ladder is the hypotenuse of a right triangle, and the distance we want to find ("x") is one of the legs. The other leg is the height of the ladder on the wall, which we don't need to know in order to solve the problem.

We can use the trigonometric function for the sine of an angle to relate the angle of elevation to the distance "x" and the length of the ladder:

sin(40°) = x / 20

To solve for "x", we can multiply both sides by 20:

20 sin(40°) = x

Using a calculator, we can evaluate the sine of 40 degrees to get:

20 sin(40°) ≈ 12.85

So the distance of the foot of the ladder from the wall is approximately 12.85 meters, rounded to two decimal places.

The correct function equivalent to the original function V = 40,000(0.64) is V = 40,000(0.64)ⁿ.

Percent is a way of expressing a fraction or a decimal as a fraction of 100. It is denoted by the symbol "%". For example, the fraction 3/4 can be written as 75% and the decimal 0.5 can be written as 50%. Percentages are commonly used in everyday life to express values such as discounts, interest rates, and success rates.

Here,

The original function V = 40,000(0.64) models the value, V, of a $40,000 car after t years, where the car depreciates at a rate of 36% per year. The expression 0.64 represents the remaining value of the car after one year, which is obtained by subtracting 36% from 100% (i.e., 100% - 36% = 64%).

To find an equivalent function, we need to simplify 0.64. We can write 0.64 as 0.8², since 0.8 multiplied by itself gives 0.64.

So, V = 40,000(0.64) is equivalent to V = 40,000(0.8²), which simplifies to V = 40,000(0.64). This means that both expressions represent the same function that models the value of the car after t years, given that the car depreciates at a rate of 36% per year.

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The percent of the bottle of apple juice is 40%.

An element of a whole number is a fraction. There are many different kinds of fractions. Three-quarters, for instance, is a fraction.

A decimal point separates a fractional component from an integral component to form a decimal. In the decimal 2.5, for instance, the fractional portion comes after the decimal point and is followed by the integer 2, which is put before it.

Per cent simply means one in a hundred. Hence, it is a number with a fractional representation that has a denominator of 100. As a result, fractions of a whole number are also described using percentages.

Given that, 80-ounce bottle of apple juice contains 32 ounces of water.

Thus,

% of water = (32 / 80) * 100 = 40%

Hence, the percent of the bottle of apple juice is 40%.

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Answer: -1/2

Step-by-step explanation:

To find the slope: rise/run

starting from the point (0,2), you go up 1 to reach the line

Therefore 1 is your rise.

Then go right 2 units and you hit the line again.

Therefore the slope is 1/2.

But the linear line is going downwards, so it is a negative slope

Therefore your final answer is:

-1/2

Answer: 14 dollars.

Step-by-step explanation:

You (1) + your four friends (4) is 5.

5 times a number is 70.

5x=70

70/5=14

x=14

Therefore, the value of hydraulic conductivity using the Thiem equation is approximately 10.67 ft/day.

An equation is a mathematical statement that indicates that two expressions are equal. It typically consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The expressions on both sides of the equation are separated by an equal sign "=" which means that the two expressions have the same value. Equations are used to model relationships between variables, solve problems, and make predictions in various fields of science, engineering, economics, and mathematics. They are an essential tool in algebra, calculus, and other branches of mathematics.

Here,

To compute the value of hydraulic conductivity using the Thiem equation, we need to substitute the given values into the equation:

K = (Q / π) * ((h2 - h1) / log(r2/r1))

where:

Q = pumping rate = 78,000 ft³/d

h1 = head at the pumping well = 260 ft ASL

h2 = head at the observation well 385 ft away = 291 ft ASL

r1 = distance from the pumping well to the observation well 125 ft away = 125 ft

r2 = distance from the pumping well to the observation well 385 ft away = 385 ft

Substituting the values, we get:

K = (78,000 / π) * ((291 - 260) / log(385/125))

K = 10.67 ft/day

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