suppose the total area is 132 square units, b=5, d=4, and the area I is a square. Find the missing lengths and areas.

To find \(\( f'(x) \)\) using the definition of the derivative, we have:

\(\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]\)

Given that \(\( f(x) = x + \frac{1}{x} \)\) , let's substitute this into the definition:

\(\[ f'(x) = \lim_{h \to 0} \frac{(x+h) + \frac{1}{x+h} - (x + \frac{1}{x})}{h} \]\)

Simplifying the numerator, we get:

\(\[ f'(x) = \lim_{h \to 0} \frac{x + h + \frac{1}{x+h} - x - \frac{1}{x}}{h} \]\)

Combining like terms in the numerator, we have:

\(\[ f'(x) = \lim_{h \to 0} \frac{h + \frac{1}{x+h} - \frac{1}{x}}{h} \]\)

Now, let's find a common denominator for the fractions:

\(\[ f'(x) = \lim_{h \to 0} \frac{hx(x+h) + (x+h) - hx}{hx(x+h)} \]\)

Simplifying further, we get:

\(\[ f'(x) = \lim_{h \to 0} \frac{hx^2 + hx^2 + hx^2 + h^2x + x^2 + hx - hx}{hx(x+h)} \]\)

Combining like terms, we obtain:

\(\[ f'(x) = \lim_{h \to 0} \frac{2hx^2 + h^2x + x^2}{hx(x+h)} \]\)

Factoring out an \(\( h \)\) from the numerator, we have:

\(\[ f'(x) = \lim_{h \to 0} \frac{h(2x^2 + hx + x^2)}{hx(x+h)} \]\)

Canceling out the \(\( h \)\) terms, we get:

\(\[ f'(x) = \lim_{h \to 0} \frac{2x^2 + hx + x^2}{x(x+h)} \]\)

Simplifying further, we have:

\(\[ f'(x) = \lim_{h \to 0} \frac{3x^2 + hx}{x(x+h)} \]\)

Taking the limit as \(\( h \)\) approaches 0, we obtain:

\(\[ f'(x) = \frac{3x^2}{x^2} \]\)

Simplifying, we find:

\(\[ f'(x) = 3 \]\)

Therefore, the derivative of  \(\( f(x) = x + \frac{1}{x} \) is \( f'(x) = 3 \).\)

Now, let's specify the domains of \(\( f(x) \) and \( f'(x) \)\). The function \(\( f(x) = x + \frac{1}{x} \)\) is defined for all \(\( x \neq 0 \)\), since division by 0 is undefined. Therefore, the domain of \(\( f(x) \) is \( (-\infty, 0) \cup (0, \infty) \)\).

The derivative \(\( f'(x) = 3 \)\)  is a constant function, which means it is defined for all real numbers. Therefore, the domain of \(\( f'(x) \) is \( (-\infty, \infty) \).\)

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Answer:

(f - g)(x) = -x² + 2x + 8

Step-by-step explanation:

Step 1: Plug in variables

2x + 1 - (x² - 7)

Step 2: Distribute the negative

2x + 1 - x² + 7

Step 3: Combine like terms

-x² + 2x + 8

Answer:

  -x^2 +2x +8

Step-by-step explanation:

F(x) = 2x+1

g(x) =x^2-7

(f-g) (x) =2x+1  - ( x^2 -7)

   Distribute the minus sign

           = 2x+1 -x^2 +7

           -x^2 +2x +8

2n^3 + 3n + 10 can be written as a polynomial of the form Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3).

To prove that the expression 2n^3 + 3n + 10 is in the set Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3), we need to show that the expression can be written in the form a(n^3) for some constant "a".

Let's start by factoring out the common factor of n^3 from each term:

2n^3 + 3n + 10 = n^3(2 + 3/n^2 + 10/n^3)

Now, let's rewrite the expression as a single term multiplied by n^3:

2n^3 + 3n + 10 = (2 + 3/n^2 + 10/n^3)n^3

Simplifying the expression inside the parentheses:

= (2n^3 + 3n^2 + 10n^3)/n^3

= (12n^3 + 3n^2)/n^3

= 12 + 3/n

ow, we can see that the expression can be written in the form a(n^3), where a = 12 and n^3 = 3/n.

Therefore, we have shown that 2n^3 + 3n + 10 can be written as a polynomial of the form Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3).

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Answer:

\(\dfrac{\sqrt{2}\sqrt[4]{3}}{3}\)

Step-by-step explanation:

Given radical expression:

\(\dfrac{\sqrt[4]{4}}{\sqrt[4]{27}}\)

Rewrite 4 as 2² and 27 as 3³:

\(\implies \dfrac{\sqrt[4]{2^2}}{\sqrt[4]{3^3}}\)

\(\textsf{Apply the exponent rule:} \quad \sqrt[n]{a}=a^{\frac{1}{n}}\)

\(\implies \dfrac{\left(2^2\right)^{\frac{1}{4}}}{\left(3^3\right)^{\frac{1}{4}}}\)

\(\textsf{Apply the exponent rule:} \quad (a^b)^c=a^{bc}\)

\(\implies \dfrac{2^{\frac{2}{4}}}{3^{\frac{3}{4}}}\)

Simplify the numerator:

\(\implies \dfrac{2^{\frac{1}{2}}}{3^{\frac{3}{4}}}\)

\(\implies \dfrac{\sqrt{2}}{3^{\frac{3}{4}}}\)

Multiply the numerator and denominator by \(3^{\frac{1}{4}}\):

\(\implies \dfrac{\sqrt{2}\cdot 3^{\frac{1}{4}}}{3^{\frac{3}{4}}\cdot 3^{\frac{1}{4}}}\)

\(\textsf{Apply the exponent rule:} \quad a^b \cdot a^c=a^{b+c}\)

\(\implies \dfrac{\sqrt{2}\cdot 3^{\frac{1}{4}}}{3^{\frac{3}{4}+\frac{1}{4}}}\)

\(\implies \dfrac{\sqrt{2}\cdot 3^{\frac{1}{4}}}{3^1}\)

\(\implies \dfrac{\sqrt{2}\cdot 3^{\frac{1}{4}}}{3}\)

\(\textsf{Apply the exponent rule:} \quad a^{\frac{1}{n}}=\sqrt[n]{a}\)

\(\implies \dfrac{\sqrt{2}\sqrt[4]{3}}{3}\)

Answer:

The vertex is at (5, -9)

Step-by-step explanation:

The vertex is halfway between the zeros

f(x) = (x-8)(x - 2)

0  = (x-8)(x - 2)

x=8 and x=2 are the two zeros

(8+2)/2 = 10/2 = 5

The x coordinate is at 5

The y coordinate is found by substituting the x coordinate into the function

f(5) = (5-8)(5 - 2) = -3 (3) = -9

The vertex is at (5, -9)

Anaiah currently spending  $150 on these goods

*He is not spending his entire income  

*His marginal utility per dollar spent is not the same for both goods

What is unit rate?

A unit rate is the cost for only one of anything. This is expressed as a ratio with a denominator of 1. For instance, if you covered 70 yards in 10 seconds, you did so at an average speed of 7 yards per second. Although both of the ratios—70 yards in 10 seconds and 7 yards in one second—are rates, only the latter is a unit rate.

consumption of eggplant

= 20 price of eggplant

= 3

consumption of kiwi is

= 30 price of eggplant

= 3

Formula:

Spending

= price of eggplant*consumption of eggplant + price of kiwi*consumption of kiwi

= 3*20+3*30

= 150

Utility per one dollar spent on the 20th eggplant

=  160

Marginal utility per dollar spent on the 30th kiwi

= 190 utils.

Note that y 160<190

Currently spending= 150

& budget = 180

150<180

According theory of choice consumer need the maximize the utility so answer is

Answer -:

*He is not spending his entire income

*His marginal utility per dollar spent is not the same for both goods

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Answer:

Step-the base is equal to length * width.

the formula for the volume of a rectangular prism becomes length * width * height.

the ratio of the volume of a three sided figure is the ratio of their corresponding sides cubed.

so if the ratio of their volumes is 125/64, then the ratio of their corresponding sides id 5/4 because 5^3 = 125 and 4^3 = 64.

since the base area is equal to length * widfth, then the ratio of the areas becomes 5^2 / 4^2 which is equal to 25/16.

the ratio of the base area becomes 25/16.

to confirm, use an example:

original prism has dimensions as shown below:

length = 20

width = 40

height = 60

volume = 20 * 40 * 60 = 48000

base area = 20 * 40 = 800

prims that has sides in a ratio of 5/4 to original prism is shown below:

length = 20 * 5/4 = 25

width = 40 * 5/4 = 50

height = 60 * 5/4 = 75

volume = 25 * 50 * 75 = 93750

base area = 25 * 50 = 1250

ratio of the volumes = 93750 / 48000 = 1.953125

ratio of the base areas = 1250 / 800 = 1.5625

125/64 = 1.953125

25/16 = 1.5625

the formulas work.

the ratio of the volume is 125/64

the ratio of the base area is 25/16by-step explanation:

volume = 20 * 40 * 60 = 48000

base area = 20 * 40 = 800

prims that has sides in a ratio of 5/4 to original prism is shown below:

length = 20 * 5/4 = 25

width = 40 * 5/4 = 50

height = 60 * 5/4 = 75

volume = 25 * 50 * 75 = 93750

base area = 25 * 50 = 1250

Answer:

Part A: 0.20p or 0.2p (doesn't matter which one you use)

Part B: p + 0.2p

Part C: p + 0.2p = 1p + 0.2p =

   1.2p

Part D: increasing a quantity by 20% is the same as multiplying the quantity by 1.2

Part E: 1/4p ounces

Part F: p - 1/4p

Part G: = 3/4p

Part H: decreasing a quantity by 1/4 is the same as multiplying the quantity by 3/4

Step-by-step

Answer: c. a = 2y and b = -3

Step-by-step explanation:

To find the values of a and b in this division problem, we need to perform polynomial division. Starting with the dividend, 8y^2 – 12y + 4, we divide the leading term of the dividend, 8y^2, by the leading term of the divisor, 4y. This gives a quotient of 2y.

Next, we multiply the divisor, 4y, by 2y and subtract it from the dividend, 8y^2 – 12y + 4. This gives a remainder of -3.

So, in this division problem, the values of a and b are a = 2y and b = -3.

To write the equation in proper form, we can divide the entire equation by \(t^9\):

\[y^{(5)} - \frac{t^{-6}}{t^{-12}}y'' + 6t^{-9}y = 0\]

Simplifying further, we can multiply the equation by \(t^{12}\):

\[t^{12}y^{(5)} - t^3y'' + 6y = 0\]

Therefore, the given differential equation is:

\[t^{12}y^{(5)} - t^3y'' + 6y = 0\]

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The volume of the cube that has a surface area of 96 in.² is calculated as: 64 in.³.

A cube is a three-dimensional shape that has equal side edges of the same edge. Where a is the length of the edge of a cube, we have the following formulas:

Surface area of a cube = 6a².

Volume of a cube = a³.

Given the following:

Surface area of the cube = 96 in.².

Find the value of a, which is the length of each edge of the cube:

6a² = 96

6a²/6 = 96/6 [division property of equality]

a² = 16

Take the square root of both sides

a = √16

a = 4 in.

Find the volume of the cube:

Volume of the cube = a³ = 4³

Volume of the cube = 64 in.³

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Answer:

Answer:

y = 8*(9/4)^x

Point (1.5, 27)

Step-by-step explanation:

We can solve each unknown in separate steps. The first step is to take advantage of given point (0,8) to find the value of a. Since x is zero, b^x will just be 1, regardless of b. That makes it easy to solve for a, which is found to be 8.

Once a is known, we can use the next point (1,18) to solve for b. b is (9/4).

Once we have a and b, we have the full equation: y = 8*(9/4)^x

k is found by entering the x value and solving for y (which is k). k = 27

Answer:

The values of a and b are,

a = 5, b = 3

Step-by-step explanation:

We are given that (1,15) , and (4,405) are on the graph of the equation

y = ab^x

so,

15 = ab^(1)   (i)

405 = ab^(4)  (ii)

solving this system of equations,

dividing (ii) by (i),

405/15 = ab^(4)/ab

27 = b^(4-1)

27 = b^3

taking the cube root,

\(b = \sqrt[3]{27}\\ b = 3\)

b = 3

Putting this value in  (i),

15 = a(3)

a = 15/3

a = 5

Hence a = 5, b = 3

Answer:

x = -2

y = 10

Step-by-step explanation:

I added a photo of my solution

Answer: -3 2/5y=2 1/2x=29

Step-by-step explanation:

6x-2/5y=8 1/2x+3y=29

Separate into two equations

6x-2/5y=8 1/2x+3y

-6x            -6x

-2/5y=2 1/2x+3y

-3y              -3y

-3 2/5y =2 1/2x

Keep 29 as is

To test the null hypothesis that the daily average revenue was $675 for the coin-operated laundry, you should use a one-sample t-test.

1. State the null hypothesis (H0) and alternative hypothesis (H1):

H0: The daily average revenue is $675.

H1: The daily average revenue is not $675.

2. Determine the sample size (n), sample mean (x), and sample standard deviation (s):

n = 30 days, x = $625, and s = $75.

3. Calculate the t-score:

t = (x - μ) / (s / √n)

t = (625 - 675) / (75 / √30)

t ≈ -3.58

4. Determine the degrees of freedom (df):

df = n - 1 = 30 - 1 = 29

5. Find the critical t-value for a two-tailed test at a 0.05 significance level:

Using a t-distribution table, the critical t-value is approximately ±2.045.

6. Compare the calculated t-score to the critical t-value:

Since the calculated t-score of -3.58 is more extreme than the critical t-value of -2.045, you would reject the null hypothesis.

In conclusion, based on the one-sample t-test, there is evidence to suggest that the daily average revenue is not $675 as claimed by the current owner.

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Answer:

i think c ..............

The probability of choosing a 5 and then a 6 is 1/49

From the question, we have the following parameters that can be used in our computation:

The tiles

Where we have

Total = 7

The probability of choosing a 5 and then a 6 is

P = P(5) * P(6)

So, we have

P = 1/7 * 1/7

Evaluate

P = 1/49

Hence, the probability of choosing a 5 and then a 6 is 1/49

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Question

You randomly choose one of the tiles. Without replacing the first tile, you choose a second tile. Find the probability of the compound event. Write your answer as a fraction or percent rounded to the nearest tenth. The probability of choosing a 5 and then a 6

Answer:

2,448

Step-by-step explanation:

Answer:

Step-by-step explanation:

without using a calculator

8(306)

8(300 + 6)

2400 + 48 = 2448

the probability that a student scored greater than 65 is 0.2389.

The probability that a student scored greater than 65 can be calculated using the standard normal distribution and the z-score. The z-score represents the number of standard deviations a value is from the mean, and can be calculated as follows:

z = (x - mean) / standard deviation

Where x is the score of interest (65)

Mean is the average score of the students (70)

Standard deviation is the standard deviation of the scores (7).

Plugging in the values, we get

z = (65 - 70) / 7 = -0.714.

Using a standard normal distribution table, we can find the probability that a student scored greater than 65 by finding the area to the right of the z-score. The probability of a student scoring greater than 65 is approximately 0.2389, or 23.89%.

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Answer:

C

Step-by-step explanation:

Answer:

proportional. for every 1 unit to the right it travels 5 units up.

Answer:

Proportional

Step-by-step explanation:

It's proportional because it's a straight line,it crosses the orgin (0,0) , and because the C.O.P (constant of proportionality) is 5.

The area of the regular polygons with 12 sides(dodecagon) and 5 sides (pentagon) are 389.06 in² and 19.87 in² respectively.

Area of regular polygon = 1/2 × apothem × perimeter

perimeter = (s)side length of octagon × (n)number of side.

apothem = s/[2tan(180/n)].

11 = s/[2tan(180/12)]

s = 11 × 2tan15

s = 5.8949

perimeter = 5.8949 × 12 = 70.7388

Area of dodecagon = 1/2 × 11 × 70.7388

Area of dodecagon = 389.0634 in²

Area of pentagon = 1/2 × 5.23 × 7.6

Area of pentagon = 19.874 in²

Therefore, the area of the regular polygons with 12 sides(dodecagon) and 5 sides (pentagon) are 389.06 in² and 19.87 in² respectively.

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The statement that is true about the number of hours of daylight is: "the number of hours of daylight varies depending on the latitude of the observer."

Hence, option (a) is the correct choice.

Day length will be roughly 12 hours at latitude 0° (the Equator). The Equator region has a steady 12 hour day light all year.

Day duration can be observed to extend to 24 hours or reduce to zero if latitude increases to 80° (polar circles - north or south) (depending on time of year).

The quantity of daylight received by an observer is determined by their latitude and the season.

The number of hours of daylight changes the greatest near the equator, whereas it varies the least towards higher latitudes.

The number of daylight hours covers the hours between dawn and twilight as well as the hours from sunrise to sunset.

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Answer:

Step-by-step explanation:

a). Given equation is y = 3x - 4

   Table for the input-output values is,

   x          -1           0          1            2          3

   y          -7          -4         -1           2          5

Now we can plot these points on the graph (graph attached).

b). Equation is y = -2x + 3

   Table for the input-output values will be,

  x       -1          0         1        2       3

  y       5          3         1        -1       -3

Answer:

29,028 ft.

Step-by-step:

Consider the following:

Mt. Everest - x

Equation:

We are given that Mt. McKinley is 20,321 ft in elevation, and that it is 8,707 ft. lower than Mt. Everest.

When writing this numerically:

20,321 = x - 8,707

Solve for x:

20,321 = x

+8,707     + 8,707

-------------------------

x = 29,028

Mt. Everest is 29,028 ft. in elevation.

The final solution of the integral is ∫2 /1 + (1/x² - 4/x³)dx = -4ln|x| - 1/x - (5/16)x⁻² + C

To evaluate the integral ∫2 /1 + (1/x² - 4/x³)dx, we can use the partial fraction decomposition method.

First, we can factor the denominator as a common denominator:

1 + (1/x² - 4/x³) = (x³ + x - 4)/(x³ x²)

Next, we can decompose the fraction into partial fractions by finding constants A, B, and C such that:

(x³ + x - 4)/(x³ x²) = A/x + B/x² + C/(x³)

Multiplying both sides by the common denominator x³ x² and simplifying, we get:

x³ + x - 4 = A(x²) + B(x) + C(x³)

Setting x = 0, we can solve for A and get A = -4.

Similarly, setting x = 1, we can solve for B and get B = 1.

Finally, setting x = -1, we can solve for C and get C = -5/4.

Therefore, the partial fraction decomposition is:

(x³ + x - 4)/(x³ x²) = (-4/x) + (1/x²) - (5/4)/(x³)

Using this decomposition, we can integrate the function term by term.

∫(-4/x)dx = -4ln|x| + C₁

∫(1/x²)dx = -1/x + C₂

∫(-5/4x³)dx = (-5/16)x⁻²  + C₃

Therefore, the final solution of the integral is:

∫2 /1 + (1/x² - 4/x³)dx = -4ln|x| - 1/x - (5/16)x⁻² + C

where C is the constant of integration.

In summary, to evaluate a complex integral like the one above, we can use the partial fraction decomposition method to simplify the function and break it down into partial fractions. Then, we can integrate each term separately and sum them up, including the constant of integration.

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Answer:

18

Step-by-step explanation:

9, 18, 27, 36, 45, 54, 63, 72, 81, 90

6, 12, 18, 24, 30, 36, 42, 48, 54, 60

The measure of the base of the triangle is 10 millimetres

From the question, we have the following parameters that can be used in our computation:

Area of the triangle = 80 square millimetres

Height of the triangle = 16 millimetres

Using the above as a guide, we have the following:

The base of the triangle = 2 * Area/Height

Substitute the known values in the above equation, so, we have the following representation

The base of the triangle = 2 * 80/16

Evaluate

The base of the triangle = 10

Hence, the base is 10 millimetres

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Complete question

The area of the triangle is 80 square millimeters and the height of the triangle is 16 millimetres.

Calculate the length of the base

Answer:

Option C is correct.

The standard deviation of B is greater.

Step-by-step explanation:

Distribution A is given

A = {91, 94, 89, 93, 87, 92, 86, 87, 92, 91}

Then distribution B is all of distribution A including the percentage for the package where the least kernels popped

B = {74, 91, 94, 89, 93, 87, 92, 86, 87, 92, 91}

Taking the statements one at a time

- The mean of A is the same as the mean of B.

Mean = (sum of variables)/(sample size)

It is evident that most of the percentages in distribution A hover around a consistent spot, hence, the mean too will land around that consistent spot. The mean of distribution A as calculated = (902/10) = 90.2

Mean of distribution B should hover around that spot too, but a new addition of 74 is expected to bring the mean down.

Mean of distribution B = (976/11) = 88 73

Hence, statement 1 is incorrect as the mean of A isn't the same as the mean of B.

- Including the percentage for the package where the least kernels popped causes the center of the data to increase.

The center of the data is synonymous with the mean and it is evident that this centre doesn't increase after including the percentage for the package where the least kernels popped causes the center of the data to increase, rather, it drops from 90.2 to 88.73.

Hence, this statement is false.

- The standard deviation of B is greater

Standard deviation measures how far away from the mean the variables of the distribution are. And since we've established that the variables of distribution A hover around a spot, hence, the stamdard deviation is expected to be small. But distribution B has all of the variables of distribution A plus a value that is very far away from the Mean (74), hence, it is easy to see that distribution B will have the greater standard deviation. Let us now calculate.

Standard deviation = σ = √[Σ(x - xbar)²/N]

x = each variable

xbar = mean

N = Sample size

Standard deviation for distribution A = 2.638

Standard deviation for distribution B = 5.293

Hence we can conclude that this statement is true and the standard deviation of distibution B is greater.

Hope this Helps!!!

Answer:

B. 15 days

Step-by-step explanation:

A=10=1/10 of the job

B=12=1/12 of the job

A+B+C=4=1/4 of the job

A+B

=1/10 + 1/12

=10+12/120

=22/120

=11/60

(A+B+C) - (A+B)

=1/4 - 11/60

=5 - 11/60

=4/60

=1/15

It would take C alone 15 days to finish the job