Identify each point as a solution of the system or not a solution of the system.

Options (mark each one below): "Solution" or "Not a solution".

(0, 4)

(-2, 4)

(0, 5)

(–2, 7)

(–4, 1)

(–1, 1)

(–1.5, 3.5)

Thank you in advance, I appreciate the help.

Set up an equation based on the information given

[tex]89 + 89 + x + x = 328[/tex]

Combine like terms

[tex]89 + 89 = 178[/tex]

[tex]x + x = 2x[/tex]

[tex]2x + 178 = 328[/tex]

Solve

[tex]2x + 178 = 328[/tex]

[tex]328 - 178 = 150[/tex]

[tex]2x = 150[/tex]

[tex]x = 75[/tex]

Answer

The width of the rectangular field is 75 yards.

Answer:

Minimum Unit Cost = $14,362

Step-by-step explanation:

The standard form of a quadratic is given by:

ax^2 + bx + c

So for our function, we can say,

a = 0.6

b = -108

c = 19,222

We can find the vertex (x-coordinate where minimum value occurs) by the formula -b/2a

So,

-(-108)/2(0.6) = 108/1.2 = 90

Plugging this value into original function would give us the minimum (unit cost):

[tex]c(x)=0.6x^2-108x+19,222\\c(90)=0.6(90)^2-108(90)+19,222\\=14,362[/tex]

Answer:

The minimum unit cost is 14,362

Step-by-step explanation:

The minimum unit cost is given by a quadratic equation. Therefore the minimum value is at its vertex

For a quadratic function of the form

[tex]ax ^ 2 + bx + c[/tex]

the x coordinate of the vertex is

[tex]x=-\frac{b}{2a}[/tex]

In this case the equation is: [tex]c(x) = 0.6x^2-108x+19,222[/tex]

Then

[tex]a= 0.6\\b=-108\\c=19,222[/tex]

Therefore the x coordinate of the vertex is:

[tex]x=-\frac{(-108)}{2(0.6)}[/tex]

[tex]x=90[/tex]

Finally the minimum unit cost is:

[tex]c(90)=0.6(90)^2-108(90)+19,222\\\\c(90)=14,362[/tex]

Answer: First option.

Step-by-step explanation:

You know that the meaning of the symbol of the inequality "<" is: Less than.

So, you can check each option to find the number that could be a length of this piece of wood.

Given [tex]w<5[/tex], you can substitute each number given in the options into this inequality. Then:

[tex]1)\ w<5\\\\4.5<5\ (This\ is\ true)[/tex]

[tex]2)\ w<5\\\\6<5 (This\ is\ not\ true)[/tex]

[tex]3)\ w<5\\\\11.3<5\ (This\ is\ not\ true)[/tex]

[tex]4)\ w<5\\\\13<5\ (This\ is\ not\ true)[/tex]

Therefore, a lenght of the piece of wood could be 4.5

The equation of the tangent to the curve at the point corresponding to the given values of the parametric equations given is;

y - 1 = ¹/₁₀(x - 1)

We are given;

x = cos θ + sin(10θ)

y = sin θ + cos(10θ)

Since we want to find equation of tangent, let us first differentiate with respect to θ. Thus;

dx/dθ = -sin θ + 10cos (10θ)

Similarly;

dy/dθ = cos θ - 10sin(10θ)

To get the tangent dy/dx, we will divide dy/dθ by dx/dθ to get;

(dy/dθ)/(dx/dθ) = dy/dx =  (cos θ - 10sin(10θ))/(-sin θ + 10cos(10θ))

To get the tangent, we will put the angle to be equal to zero.

Thus, at θ = 0, we have;

dy/dx = (cos 0 - 10sin 0)/(-sin 0 + 10cos 0)

dy/dx = 1/10

Also, at θ = 0, we can get the x-value and y-value of the parametric functions.

Thus;

x = cos 0 + sin 0

x = 1 + 0

x = 1

y = sin 0 + cos 0

y = 0 + 1

y = 1

Thus, the equation of the tangent line to the curve in point slope form gives us;

y - 1 = ¹/₁₀(x - 1)

Read more at; https://brainly.com/question/13388803

Final answer:

To find the tangent line to the curve defined by the parametric equations at θ = 0, we compute the derivatives of both x and y with respect to θ, leading to a slope of 1/10. By evaluating the original parametric equations at θ = 0, we find that the tangent passes through (1, 1), resulting in the equation y - 1 = 1/10(x - 1).

Explanation:

To find an equation of the tangent to the given parametric curve at θ = 0, we first need the parametric equations given by x = cos(θ) + sin(10θ) and y = sin(θ) + cos(10θ). To find the slope of the tangent, we compute the derivatives δy/δx = (δy/δθ)/(δx/δθ) at θ = 0.

Computing the derivatives, δx/δθ = -sin(θ) + 10cos(10θ) and δy/δθ = cos(θ) - 10sin(10θ), and plugging in θ = 0, we get δx/δθ = 10 and δy/δθ = 1. Hence, the slope is 1/10. Evaluating the functions at θ = 0 gives x = 1 and y = 1. Thus, the tangent line equation at θ = 0 is y - y_0 = m(x - x_0), which simplifies to y - 1 = 1/10(x - 1).

Answer:

7 cents/mile

Step-by-step explanation:

You are looking for a unit rate of cents per mile.

Change the dollar amount to cents, and divide by the number of miles.

$13.08 * (100 cents)/$ = 1308 cents

(1308 cents)/(183 miles) = 7.001 cents/mile

Answer:

[tex]\displaystyle \lim_{x \to 0} \Big( x^3 + \frac{1}{x^5} \Big) = \text{und} \text{efined}[/tex]

General Formulas and Concepts:

Calculus

Limits

Limit Rule [Variable Direct Substitution]:                                                             [tex]\displaystyle \lim_{x \to c} x = c[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \lim_{x \to 0} \Big( x^3 + \frac{1}{x^5} \Big)[/tex]

Step 2: Evaluate

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits

Answer: 24, 44, and 66 [tex]cm^{2}[/tex]

Step-by-step explanation:

Check all of the possible combinations of the faces:

6 × 4 = 24

11 × 4 = 44

11 × 6 = 66

So the answers are 24, 44, and 66 [tex]cm^{2}[/tex]

Answer:

option B 24 cm² option C 44 cm² and option E 66 cm² are the correct options

Step-by-step explanation:

A rectangular prism has 6 faces, in which opposite faces are always similar.

So a rectangular prism has 3 different faces

(1) From the figure given

for face (1)  Area = 11 × 4

                           = 44 cm²

For face (2) Area = 4 × 6

                            = 24 cm²

For face (3) Area = 11 × 6

                             =66 cm²

Now we can say that option B 24 cm² option C 44 cm² and option E 66 cm² are the correct options

Answer: Option A

Step-by-step explanation:

Given the parent function [tex]f(x)=x^5[/tex], it can be transformated:

If  [tex]f(x)=x^5-k[/tex], then the function is shifted k units down.

If  [tex]f(x)=a(x^5)[/tex] and [tex]a > 1[/tex]  it is vertically stretched it, but if [tex]0 < a < 1[/tex] it is vertically compressesd.

If  [tex]f(x)=-(x^5)[/tex], then the function is reflected over the x-axis.

Then, if the function given is reflected over the x-axis, it is vertically streteched by a factor o 2 and it is shifted down 1 units, the function that results after this transformations is:

[tex]g(x)=-2(x^5)-1[/tex]

[tex]g(x)=-2x^5-1[/tex]

Answer:

4185

Step-by-step explanation:

A culture of bacteria grows exponentially according to the following general exponential growth function;

[tex]P_{t}=P_{0}e^{kt}[/tex]

where;

p(t) is the population at any given time t.

p(0) is the initial population

k is the growth constant

From the information given we have;

p(0) = 1500

at t = 5, p(t) = 2300; p(5) = 2300

We shall use this information to determine the value of k;

[tex]2300=1500e^{5k}[/tex]

Divide both sides by 1500;

[tex]\frac{23}{15}=e^{5k}\\\\ln(\frac{23}{15})=5k\\\\k=0.08549[/tex]

Therefore, the population of the bacteria at any time t is given by;

[tex]P_{t}=1500e^{0.08549t}\\\\P(12)=1500e^{0.08549(12)}=4184.3[/tex]

Answer:

  about 2.33

Step-by-step explanation:

The value can be found from a probability table, any of several web sites, your graphing calculator, most spreadsheet programs, or any of several phone or tablet apps.

A web site result is shown below. (I have had trouble in the past reconciling its results with other sources.) One of my phone apps gives the z-value as about ...

  2.26347874

which is in agreement with my graphing calculator.

Answer:

[tex]Z = 2.325[/tex].

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Find the z score that corresponds to P99, the 99th percentile of a standard normal distribution curve.

This is the value of Z when X has a pvalue of 0.99. This is between 2.32 and 2.33, so the answer is [tex]Z = 2.325[/tex].

Answer:

g(x) = x^2 - 11x + 26

Step-by-step explanation:

In translation of functions, adding a constant to the domain values (x) of a function will move the graph to the left, while subtracting from the input of the function will move the graph to the right.

Given the function;

f(x) = x2 - 3x - 2

a shift 4 units to the right implies that we shall be subtracting the constant 4 from the x values of the function;

g(x) = f(x-4)

g(x) = (x - 4)^2 - 3(x - 4) -2

g(x) = x^2 - 8x + 16 - 3x + 12 - 2

g(x) = x^2 - 11x + 26

Answer:

6t+3

Step-by-step explanation:

If t represents the number of tens, then 6t is six times the number of tens. 3 more than that is ...

6t+3

Answer:

6t + 3

Step-by-step explanation:

Given: Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens

To Find: Let t represent the number of tens. Write an expression for the number of fives.

Solution:

Total number of ten bills are = [tex]\text{t}[/tex]

As given in question,

The number of fives is three more than six times the number of tens

therefore

total number of fives are

                                          =[tex]6\text{t}+3[/tex]

here,  t represents total number of $5 and $10 bills Jeannette has in her wallet

Final expression for total number of [tex]\$5[/tex] bills is [tex]6\text{t}+3[/tex]

Answer:

20,309 people more people per sq mile.

Step-by-step explanation:

First step is to calculate the population density of both cities, then we'll be able to answer the question.

We're looking for a number of people per sq mile... so we'll divide the population by the area.

New York City: 8.54 million people on 302.6 sq miles

DensityNYC = 8,540,000 / 302.6 =  28,222 persons/sq mile

Los Angeles: 3.98 million people on 503 sq miles

DensityLA = 3,980,000 / 503 = 7,913 persons/sq mile

Then we do the difference...  28,222 - 7,913 = 20,309 people more people per sq mile.

If we were to make the ratio, we'd get 3.57, 3.57 more people in NYC per sq mile compare to LA.

Final answer:

New York City has approximately 20,309 more people per square mile than Los Angeles when rounded to the nearest person, based on their population densities.

Explanation:

To calculate how many more people per square mile live in New York City versus Los Angeles, we need to find the population density for each city and then subtract Los Angeles's density from New York City's density.

New York City's population density: 8.54 million people ÷ 302.6 square miles = approximately 28,220 people per square mile.

Los Angeles's population density: 3.98 million people ÷ 503 square miles = approximately 7,911 people per square mile.

The difference in population density: 28,220 people per square mile (NYC) - 7,911 people per square mile (LA) = approximately 20,309 people per square mile.Therefore, about 20,309 more people per square mile live in New York City than in Los Angeles, when rounded to the nearest person.

Answer:

  see below

Step-by-step explanation:

The slopes of parallel lines are the same. The slopes of perpendicular lines are negative reciprocals of each other, hence their product is -1.

___

For the most part, the concept of adding slopes of lines does not relate to parallel or perpendicular lines in any way.

Answer:

c

Step-by-step explanation:

Answer:

=> 6s

Step-by-step explanation:

Given terms

24s^3  ,12s^4  and 18s

GCF consists of the common factors from all the terms whose GCF has to be found.

In order to find GCF, factors of each term has to be made:

The factors of 24s^3:

24s^3=2*2*2*3*s*s*s

The factors of 12s^4:

12s^4=2*2*3*s*s*s*s

The factors of 18s:

18s=2*3*3*s

The common factors are(written in bold):

GCF=2*3*s

=6s

So the GCF is 6s ..  

Answer:

6s

Step-by-step explanation:

Answer:

  16 1/4 years

Step-by-step explanation:

Assuming interest is compounded annually, the account balance (A) after t years will be ...

  A = P(1 +r)^t

  3200 = 1700·1.0397^t

  log(3200) = log(1700) +t·log(1.0397)

  t = (log(3200) -log(1700))/log(1.0397) ≈ 16.247

The account will reach a balance of $3200 after about 16 1/4 years.

___

You may be asked to round your answer to the nearest integer or tenth. We leave that exercise to the student.

Based on the pattern table, the value of a is 1/64.

Given:

Find:

Solution:

The negative exponent rule tells us that a number with a negative exponent should be put to the denominator, and vice versa.

So, [tex]2^{-6} = \frac{1}{2^{6} } = \frac{1}{64}[/tex]

As, [tex]2^{6} = 64[/tex].

So, a = 1/64

Hence, the value of a is 1/64

To learn more about patterns, refer to:

https://brainly.com/question/854376

#SPJ2

Step-by-step explanation:

Remember that in a linear function of the form [tex]f(x)=mx+b[/tex], [tex]m[/tex] is the slope and [tex]b[/tex] is the why intercept.

Part A. Since [tex]g(x)=2x+6[/tex], its slope is 2 and its y-intercept is 6

Now, to find the slope of [tex]f(x)[/tex] we are using the slope formula:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

where

[tex]m[/tex] is the slope

[tex](x_1,y_1)[/tex] are the coordinates of the first point

[tex](x_2,y_2)[/tex] are the coordinates of the second point

From the table the first point is (-1, -12) and the second point is (0, -6)

Replacing values:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m=\frac{-6--(12)}{0-(-1)}[/tex]

[tex]m=\frac{-6+12}{0+1}[/tex]

[tex]m=6[/tex]

The slope of f(x) is bigger than the slope of g(x), which means the line represented by f(x) is stepper than the line represented by g(x).

Part B. To find the y-intercept of f(x) we are taking advantage of the fact that the y-intercept of a linear function occurs when x = 0, so we just need to look in the table for the value of f(x) when x = 0. From the table [tex]f(x)=-6[/tex] when [tex]x=0[/tex]; therefore the y-intercept of [tex]f(x)[/tex] is -6.

We already know that the y-intercept of g(x) is 2. Since 2 is bigger than -6, function g(x) has a greater y-intercept.

Answer:

x = 1.5

Step-by-step explanation:

The left side of the equation is graphed as a straight line with a slope of -5/3 and a y-intercept of +7. The right side of the equation is graphed as a horizontal line at y = 4.5. The point of intersection of these lines has the x-coordinate of the solution: x = 1.5.

Answer:

  D. slope = 3 and y-intercept = -3

Step-by-step explanation:

The equation can be rearranged by adding the opposite of the term with parentheses:

  y = 3(x -1)

Expanding this to slope-intercept form gives ...

  y = 3x -3 . . . . . . . . slope = 3, y-intercept = -3

_____

Slope-intercept form is ...

  y = mx +b . . . . . . . . slope = m, y-intercept = b

Answer:

365

Step-by-step explanation:

ANSWER

B. 3 ft by 8 ft

EXPLANATION

The area is given as 24 square feet.

This implies that,

[tex]l \times w = 24[/tex]

The perimeter of the rectangular field is given as 22 feet.

This implies that,

[tex]2(l + w) = 22[/tex]

Or

[tex]l + w = 11[/tex]

We make w the subject in this last equation and put it inside the first equation.

[tex]w = 11 - l[/tex]

When we substitute into the first equation we get;

[tex]l(11 - l) = 24[/tex]

[tex]11l - {l}^{2} = 24[/tex]

This implies that,

[tex] {l}^{2} - 11l + 24 = 0[/tex]

[tex](l - 3)(l - 8) = 24[/tex]

[tex]l = 3 \: or \: 8[/tex]

When l=3, w=24

Therefore the dimension is 3 ft by 8 ft

Answer:

The correct answer is option B.  3 ft  by 8 ft

Step-by-step explanation:

Points to remember

Area of rectangle = length * breadth

Perimeter of rectangle = 2(Length + Breadth)

It is given that, The area of a rectangular flower bed is 24 square feet. The perimeter of the same flower bed is 22 feet

To find the correct option

1). Check option A

Area = 2 * 12 = 24

Perimeter = 2( 2 + 12 ) = 28

False

2) Check option B

Area = 3 * 8 = 24

Perimeter = 2(3  + 8 ) = 22

True

3). Check option C

Area = 3 * 6 = 18

Perimeter = 2( 3 + 6 ) = 18

False

4). Check option D

Area = 4 * 6 = 24

Perimeter = 2( 4 +6 ) = 20

False

The correct answer is option B.  3 ft  by 8 ft

Final answer:

The variables in this problem represent the scores in the score matrix S. The objective for this 0-1 integer linear program is to maximize the overall score. The entire 0-1 integer linear program in standard form includes constraints to ensure that each detection is assigned to a single track and each track is assigned only once, and there are N + M constraints in total.

Explanation:

(a) In this problem, the variables represent the scores in the score matrix S. There are N detections and M tracks, so we have N rows and M columns in the score matrix.

(b) The objective for this 0-1 integer linear program is to maximize the overall score, which is the sum of the selected detections' scores.

(c) The entire 0-1 integer linear program in standard form can be defined as:

There are N + M constraints in total.

I'm unsure about this answer, but I got 108 people. I just did a simple proportion of 6/1500 = x/27000 and solved for x.

Answer:

108 people

Step-by-step explanation:

We can write a proportion to solve this problem.  Take the number of people over the living space

6 people           x people

------------    = ------------------

1500 ft^2            27000 ft^2

Using cross products

6 * 27000  = 1500 x

Divide each side by 1500

6 * 27000/1500 = 1500x/1500

108 = x

108 people  can be reasonably accommodated

Answer:

The dimensions that will minimize cost are 3.42 cm and 6.84 cm

Step-by-step explanation:

* Lets explain how to solve this problem

- We have a storage box with a square base

- The volume of the box is 80 cm³

* From the information above we can find relation between the two

 dimensions of the box

∵ The base of the box is a square with side length L cm

∵ The height of the box is H cm

∵ The volume of the box = area of its base × its height

- The base is a square and area the square = L² cm²

∴ The volume of the box = L² × H

∵ The volume of the box = 80 cm³

∴ L² × H = 80

- Lets find H in terms of L by divide both sides by L²

∴ H = 80/L² ⇒ (1)

- The cost of the top and bottom is $0.20 per cm²

- We can find the cost of top and bottom by multiplying the area of

  them by the cost per cm²

∵ The top and the bottom are squares with side length L cm

∴ The area of them = 2 × L² = 2L² cm²

∵ The cost per cm² is $0.20

∴ The cost of top and bottom = 2L² × 0.20 = 0.40L² ⇒ (2)

- Now we can find the cost of the lateral area (area of the 4 side faces)

 by multiplying the area of them by the cost per cm²

∵ The lateral area = the perimeter of its base × its height

∵ The base is a square with side length L cm

∴ The perimeter of the base = 4 × L = 4L cm

∵ The height of the box is H cm

∴ The lateral area = 4L × H

- Now lets replace H by L using equation (1)

∴ The lateral area = 4L × 80/L²

- To simplify it : 4 × 80 = 320 and L/L² = 1/L

∴ The lateral area = 320/L cm²

∵ The cost of the sides is $0.10 per cm²

∴ The cost of the lateral area = 320/L × 0.10 = 32/L ⇒ (3)

- Now lets find the total cost of the box by adding (2) and (3)

∴ The total cost (C) = 0.40L² + 32/L

* For the minimize cost we will differentiate the equation of the

  cost C with respect to the dimension L (dC/dL) and equate it

  by 0 to find the value of L which makes the cost minimum

- In differentiation we multiply the coefficient of L by its power and

 subtract 1 from the power

∵ C = 0.40L² + 32/L

- Lets change 32/L to 32L^(-1) ⇒ (we change the sign of the power by

 reciprocal it)

∴ C = 0.40L² + 32L^(-1)

-  Lets differentiate

∴ dC/dL = (0.40 × 2)L^(2 - 1) + (32 × -1)L^(-1 - 1)

∴ dC/dL = 0.80L - 32L^(-2)

- For the minimum cost put dC/dL = 0

∴ 0.80L - 32L^(-2) = 0 ⇒ add 32L^(-1) to both sides

∴ 0.80L = 32L^(-2)

- Change 32L^(-2) to 32/L² (we change the sign of the power by

 reciprocal it)

∴ 0.80L = 32/L² ⇒ use cross multiplication to solve it

∴ L³ = 32/0.80 = 40 ⇒ take ∛ for both sides

∴ L = ∛40 = 3.41995 ≅ 3.42 cm ⇒ to the nearest 2 decimal place

- Substitute this value of L in equation (1) to find H

∵ H = 80/L²

∴ H = 80/(∛40)² = 6.8399 ≅ 6.84 cm ⇒ to the nearest 2 decimal place

* The dimensions that will minimize cost are 3.42 cm and 6.84 cm

Taking the cubic root of a number is the same as raising that number to the power of 1/3.

Moreover, we have

[tex]64 = 2^6[/tex]

So, we have

[tex]\sqrt[3]{64} = \sqrt[3]{2^6} = (2^6)^{\frac{1}{3}} = 2^{6\cdot\frac{1}{3}} = 2^2 = 4 [/tex]

Answer:

4

Step-by-step explanation:

Since we see a cube root, we will attempt to rewrite 64 as a number with an exponent of 3.

[tex]\sqrt[3]{64}[/tex]

[tex]= \sqrt[3]{4^3}[/tex]

[tex]= 4 [/tex]

Answer:

about 10.06 oz.

Step-by-step explanation:

Let x represent the number of ounces of pure gold you need to add. Then the amount of gold in the mix is ...

100%·x + 75%·5 = 91.7%·(x+5)

8.3%·x = 5·16.7% . . . . . . subtract 91.7%·x +75%·5

x = 5 · 16.7/8.3 . . . . . . . . divide by the coefficient of x

x ≈ 10.06 . . . . oz

_____

Alternate solution

The amount of non-gold in the given material is 25%·5 oz = 1.25 oz. That is allowed to be 8.3% of the final mix, so the weight of the final mix will be ...

(1.25 oz)/0.083 ≈ 15.06 oz

Since that weight will include the 5 oz you already have, the amount of pure gold added must be ...

15.06 oz - 5 oz = 10.06 oz

_____

Comment on these answers

If you work directly with carats instead of percentages, you find the amount of pure gold you need to add is 10.00 ounces, double the amount you have.

Answer:

Membership decreased by an average of 6,600 people per year from Year 3 to Year 5

Step-by-step explanation:

The average rate of change from Year 3 to Year 5 will be given by the slope of the line joining the points;

(3, 96.8) and (5, 83.6)

The slope of a line given two points is calculated as;

( change in y)/( change in x)

In this case y is the number of members for a given year x.

average rate of change = (83.6-96.8)/(5-3)

                                        = -6.6

Since the number of members is given in thousands, we have;

-6,600

The negative sign implies a decrease in the number of members. Therefore, membership decreased by an average of 6,600 people per year from Year 3 to Year 5

Use Slide method for some of them, some are perfect squares:

Watch this video and try it a few times, pay attention to the signs when you are seeing which one will be negative when you multiply it out, in picture is me working out all problems requested.

Ask your brother for further help!

In the [tex]x[/tex]-[tex]y[/tex] plane, the base has equation(s)

[tex]16x^2+9y^2=144\implies y=\pm\dfrac43\sqrt{9-x^2}[/tex]

which is to say, the distance (parallel to the [tex]y[/tex]-axis) between the top and the bottom of the ellipse is

[tex]\dfrac43\sqrt{9-x^2}-\left(-\dfrac43\sqrt{9-x^2}\right)=\dfrac83\sqrt{9-x^2}[/tex]

so that at any given [tex]x[/tex], the cross-section has a hypotenuse whose length is [tex]\dfrac83\sqrt{9-x^2}[/tex].

The cross-section is an isosceles right triangle, which means the legs occur with the hypotenuse in a ratio of 1 to [tex]\sqrt2[/tex], so that the legs have length [tex]\dfrac8{3\sqrt2}\sqrt{9-x^2}[/tex]. Then the area of each cross-section is

[tex]\dfrac12\left(\dfrac8{3\sqrt2}\sqrt{9-x^2}\right)\left(\dfrac8{3\sqrt2}\sqrt{9-x^2}\right)=\dfrac{16}9(9-x^2)[/tex]

Then the volume of this solid is

[tex]\displaystyle\frac{16}9\int_{-3}^39-x^2\,\mathrm dx=\boxed{64}[/tex]

Solid [tex]\( S \)[/tex] has elliptical base[tex]\( 16x^2 + 9y^2 = 144 \)[/tex]. Triangular cross-sections yield volume [tex]\( 128 \)[/tex] cubic units.

let's break it down step by step.

1. Understanding the Solid: The solid [tex]\( S \)[/tex] has a base in the shape of an ellipse given by the equation [tex]\( 16x^2 + 9y^2 = 144 \).[/tex] The cross-sections perpendicular to the x-axis are isosceles right triangles with their hypotenuse lying on the base ellipse.

2. **Equation of the Ellipse**: To understand the shape of the base, let's rearrange the equation of the ellipse to find [tex]\( y \)[/tex]  in terms of [tex]\( x \):[/tex]

 [tex]\[ 16x^2 + 9y^2 = 144 \] \[ y^2 = \frac{144 - 16x^2}{9} \] \[ y = \pm \frac{4}{3} \sqrt{9 - x^2} \][/tex]

3. Finding the Length of the Hypotenuse: The length of the hypotenuse of each triangle is twice the value of [tex]\( y \)[/tex] at any given point on the ellipse. So, the length [tex]\( h \)[/tex] of the hypotenuse is given by:

  [tex]\[ h = \frac{8}{3} \sqrt{9 - x^2} \][/tex]

4. Area of Each Cross-Section Triangle: The area of each cross-section triangle is [tex]\( \frac{1}{2} \times \text{base} \times \text{height} \),[/tex] where the base is the same as the height. So, the area is:

[tex]\[ \text{Area} = \frac{1}{2} \times \frac{8}{3} \sqrt{9 - x^2} \times \frac{8}{3} \sqrt{9 - x^2} = \frac{32}{9} (9 - x^2) \][/tex]

5. Integrating to Find Volume: To find the volume of the solid, we integrate the area function over the interval that covers the base ellipse, which is [tex]\([-3, 3]\)[/tex] in this case.

  [tex]\[ V = \int_{-3}^{3} \frac{32}{9} (9 - x^2) \, dx \][/tex]

6. Solving the Integral: Integrating [tex]\( (9 - x^2) \)[/tex] with respect to[tex]\( x \)[/tex]  yields:

  [tex]\[ = \frac{32}{9} \int_{-3}^{3} (9 - x^2) \, dx \] \[ = \frac{32}{9} \left[ 9x - \frac{x^3}{3} \right]_{-3}^{3} \] \[ = \frac{32}{9} \left[ (27 - 9) - (-27 + 9) \right] \] \[ = \frac{32}{9} \times 36 \] \[ = \frac{1152}{9} \] \[ = 128 \][/tex]

7. Final Result: So, the volume of the solid [tex]\( S \)[/tex] is [tex]\( 128 \)[/tex] cubic units.