Order the function from the least steep slope to the steepest slope.

P = distance all around

P = 2x + 3(x)

15 = 2x + 3x

15 = 5x

15/5 = x

3 = x

The distance of one of the shorter sides is 3 inches.

The length of one of the shorter sides is 3 inches.

A trapezium is a quadrilateral with four sides where two sides are parallel to each other.

We have,

Trapezium has four sides and two parallel sides.

Now,

Let three sides be equal.

i.e x

The longer sides of the parallel sides.

= 2x

The shorter sides of the parallel sides.

= x

Now,

Perimeter of the trapezium = 15 inches

2x + x + x + x = 15

2x + 3x = 5x

5x = 15

x =3

Thus,

The length of the shorter side is 3 inches.

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Answer:

Tax = $19.04

Step-by-step explanation:

Hourly wage = $6.8/hr

hours worked = 10

Tax = 28%

[tex]total \: earnings =6.8 \times 10 = 68 \\ amount \: of \: tax = 68 \times 28\% = 68 \times \frac{28}{100} \\ = 19.04[/tex]

Answer:

[tex]v(x)=32+8(x-1)[/tex]

Step-by-step explanation:

We have been given that the number of visitors to a park is expected to follow the function [tex]v(x)=8(x-1)[/tex], where x is the number of days since opening. On the first day, there will be a ceremony with 32 people in attendance.

The total number of visitors including the ceremony would be number of people on ceremony plus people at x number of days since opening that is:

[tex]v(x)=32+8(x-1)[/tex]

Therefore, the function [tex]v(x)=32+8(x-1)[/tex] total visitors, including the ceremony.

Answer:

Proportion

Step-by-step explanation:

Proportion is just the division of the data that meets the description, between the total number of data present in the study.

For example, let's suppose that we have a tiger, a lion, a sheep, a cow and a horse, and we want to know the proportion of animals that eat meat, then, only 2 out of 5 of those eat meat, the tiger and the lion, meaning [tex]\frac{2}{5}[/tex], which would be the proportion, or 0.40

Answer:

w = 0.333

Step-by-step explanation:

In circular motion you have:

v = wr

replacing:

3 = 9w

0.333 = w

The angular speed of a point with r = 9 cm and v = 3 cm/sec is 0.33 rad/s

The angular speed (ω) is given by:

ω = v/r

Where v is the linear speed and r is the radius.

Given that v = 3 cm/sec = 0.03 m/s, radius = 9 cm = 0.09 m, hence:

ω = v/r = 0.03/0.09 = 0.33 rad/s

The angular speed of a point with r = 9 cm and v = 3 cm/sec is 0.33 rad/s

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Answer:

A. Kelley uses 2 parts blue for every 3 parts yellow.

Step-by-step explanation:

Given equation that shows the amount of blue food coloring,

[tex]b=\frac{2}{3}y[/tex]

Where,

y = amount of yellow food coloring,

If y = 2,

[tex]b=\frac{2}{3}\times 2=\frac{4}{3}[/tex]

i.e. [tex]\frac{4}{3}[/tex] parts of blue for every 2 parts yellow.

If y = 3,

[tex]b=\frac{2}{3}\times 3=2[/tex]

i.e. 2 parts of blue for every 3 parts yellow.

If y = 5,

[tex]b=\frac{2}{3}\times 5=\frac{10}{3}[/tex]

i.e. [tex]\frac{10}{3}[/tex] parts of blue for every 5 parts yellow.

Hence, OPTION A is correct.

Answer:

4

Step-by-step explanation:

Recall that for a function f(x) and for a constant k

f(x+k) represents a horizontal translation for the function f(x) by k units in the negative-x direction.

Hence f(x+k) is simply the graph of f(x) that has been moved left (negative x direction) by k units.

From the graph, we can see that g(x) = f(x+k) is simply the graph of f(x) that has been moved 4 units in the negative x-direction.

hence K is simply 4 units.

Answer:

Step-by-step explanation:

Yes I'd have to agree with @previousbrainliestperson

I'd go with solid 4

Answer:

  8 1/4 inches

Step-by-step explanation:

  (3 1/4 - 1/4) + (5 1/2 - 1/4) = 3 + 5 1/4 = 8 1/4 . . . inches

Answer:

  88 m/s

Step-by-step explanation:

The appropriate formula relating initial speed v, acceleration a, and distance d is ...

  v² = 2ad

  v² = 2(8 m/s²)(484 m) = 7744 m²/s²

Taking the square root gives ...

  v = √7744 m/s = 88 m/s

Answer:

Step-by-step explanation:

The distance formula is d=[tex]\sqrt{(x2-x1)^{2} +[tex]\sqrt{(0-0)^{2} + (12-3)^{2}}[/tex][tex]\sqrt{0 + (9)^{2}}[/tex]

[tex]\sqrt{(9)^{2}}[/tex]

[tex]\sqrt{(y2-y1)^{2} }[/tex]

[tex]\sqrt{81}[/tex] = 9

Answer: Just add then you have to multiply the main number i cant give you the anwser but i can help you∈∈∈

Step-by-step explanation:

Answer:

The given statement is true.

Step-by-step explanation:

Yes this is true.

GAAP is a collection of certain standard accounting rules for financial reporting.

Few general principles of GAAP guidelines are :

1. Principle of Regularity.

2.  Principle of Sincerity.

3. Principle of Consistency.

4. Principle of Non-Compensation.

5. Principle of Continuity.

Answer:

Option c. $284.85

Step-by-step explanation:

we know that

The amount of money Diego now has is equal to the amount of money he originally had plus deposits minus withdrawals.

so

[tex]272.79+(26.32+91.03+17.64)-(31.08+29.66+62.19)\\272.79+134.99-122.93\\\$284.85[/tex]

Answer:

x=4

Step-by-step explanation:

Because we know that PQ = RS, we can use the transitive property to replace PQ in the first equation with 29:

9x-7=29

1) Add 7 to both sides:

9x=36

2) divide by 9 on both sides:

x=4

Final answer:

To find x, set the given equations equal to each other. Simplify the equation and solve for x. The solution is x = 4.

Explanation:

To find x, we can set the given equations equal to each other:

9x - 7 = 29

Adding 7 to both sides, we get:

9x = 36

Dividing both sides by 9, we find:

x = 4

So, x is equal to 4.

Answer:

c.7,999,999

Step-by-step explanation:

The phone number is of the form ABC - XXXX

A can be any number from 2 - 9. This means number of possible values for A are 8.

The rest of the places B,C and X can be any digit from 0 - 9. This means there are 10 possible values for each of these.

Since, value to A can be assigned in 8 ways, and to the rest of the 6 positions in 10 ways, according to the fundamental rule of counting, the total number of possible phone numbers that can be formed will be equal to the product of all the individual ways:

Total possible phone numbers = 8 x 10 x 10 x 10 x 10 x 10 x 10

Since, 1 of the given number: 867-5309 is not used, the total possible phone numbers will be:

Total possible phone numbers = [tex]8 \times 10^{6} - 1 = 7999999[/tex]

Hence, option C: 7,999,999 give the correct answer.

Final answer:

The answer calculates the possible phone numbers in the (770) area code with given restrictions, ending up with 7,999 possible phone numbers.

Explanation:

To calculate how many phone numbers are possible in the (770) area code with the given restrictions, we first determine the possibilities for each digit:

A (restricted to 2-9): 8 optionsB, C, X (0-9 for each digit): 10 options each

So, the total number of possible phone numbers is: 8 (A) * 10 (B) * 10 (C) * 10 (X) = 8,000. However, we need to exclude the number 867-5309, so the final count is 8,000 - 1 = 7,999.

Answer: x=7

Step-by-step explanation:

Ok, so first put SR/ML=QR/KL (not dividing)

Then, fill in the blanks, x/5=4.2/3

Then, cross multiply leaving the equals sign there

x*3=4.2*5

Then, solve for x

3x=21

— —

3. 3

Lastly you get your answer of

X=7

Hope I helped

Answer:

The value of x is 7.

Step-by-step explanation:

Consider the provided figure.

It is given that both the pentagons are similar.

That means the ratio of the sides will be same and we need to find the value of x.

[tex]\frac{NM}{TS}=\frac{ML}{SR}[/tex]

Substitute the respective values in the above formula.

[tex]\frac{4}{5.6}=\frac{5}{x}[/tex]

[tex]4x=5\times 5.6[/tex]

[tex]x=\frac{28}{4}[/tex]

[tex]x=7[/tex]

Hence, the value of x is 7.

Answer:

The time at which the pass each other on the road is 4:00 pm

Step-by-step explanation:

The first step is to write the equations that give us the position of Emily and Anthony, these are give by:

[tex]x_A = x_{A0}+v_At[/tex]  for Anthony

[tex]x_E=x_{E0}+v_Et[/tex]     for Emily

Since Antony drives 160 miles to Albany, we can claim that the distance from Kingstown to Albany is 160 miles.

Let us set the initial position of Antony in Kingstown and consider it as the origin of our coordinate system. In this way, [tex]x_{A0}=0[/tex].

This automatically tells us that Emily initial position, in Albany, is 160 miles from our origin, hence [tex]x_{E0} = 160 miles[/tex].

Now, we need to define where to start counting the time. In this problem, it is easier to set time zero when Emily leaves. The reason for this is that now, we can say that when Emily left, Anthony was already traveling during 15 mins (remember Emily departing time was 2:15 pm and Anthony's time was 2:00 pm) and Anthony's initial position was from Emily's point of view was different from zero. We can calculate this distance as the multiplication of the time Anthony was traveling times the speed at which he was driving. This is:

[tex]x_{A0}=v_At[/tex]

being [tex]t[/tex] the 15 mins he traveled before Emily started and [tex]v[/tex] the 45 m.p.h given by the problem. We also need to convert 15 mins to hours, which gives 0.25 hours. Thus:

[tex]x_{A0}=45*0.25\\x_{A0}=11.25[/tex] miles

and the position equations are now:

[tex]x_A= 11.25 + v_At[/tex] for Anthony

[tex]x_E=160 + v_Et[/tex]

Since we are asked the time at which the pass each other on the road we need to equals their positions, [tex]x_A=x_E[/tex]:

[tex]11.25+v_At=160-v_Et[/tex]

Notice here that Emily's position is negative since she is moving towards the origin of our system, meaning in the negative direction. Solving for [tex]t[/tex]:

[tex]11.25+v_At=160-v_Et\\v_At+v_Et=160-11.25\\t(v_A+v_E)=148.75\\t = \frac{148.75}{v_A+v_E}[/tex]

Substituting the values of [tex]v_A=45[/tex] and [tex]v_E=40[/tex]:

[tex]t = \frac{148.75}{45+40}=\frac{148.75}{85}\\t=1.75 h[/tex]

What we have calculated is the time interval from where we start counting the time and remember this was set at 2:15 pm when Emily left. Since the exercise asks for the hours of the day we need to add the time interval to 2:15 pm and:

[tex]1.75 h = 1 h+45 min[/tex]

And 2:15 pm + 1 h is 3:15 pm + 45 mins is 4:00 pm which is the time at which the pass each other on the road.

Final answer:

Anthony and Emily will cross paths at 4:00 p.m. after calculating the distance covered by Anthony and the remaining distance between them when Emily starts driving, with their combined speed taken into consideration.

Explanation:

To solve this question, we need to calculate the time when Anthony and Emily will cross paths on the road, given that they are traveling towards each other from Kingstown to Albany and vice-versa. Anthony drives at 45 mph whereas Emily drives at 40 mph

Let's first find out how far apart they are when Emily starts her journey at 2:15 p.m. Since Anthony left at 2:00 p.m. and drives for 15 minutes until Emily starts her journey, we calculate the distance he has covered as:

Now, the remaining distance between them is:

The combined speed at which they're closing the distance is:

To find the time it takes for them to meet, we use the formula:

Since 1.75 hours is 1 hour and 45 minutes, they will meet at:

Therefore, Anthony and Emily will cross paths at 4:00 p.m.

The event of the machine working is [tex]A\cap B\cap C\cap D[/tex], and since the components operate independently, we have

[tex]P(A\cap B\cap C\cap D)=P(A)P(B)P(C)P(D)[/tex]

so just multiply the given probabilities together,

[tex]P(A\cap B\cap C\cap D)=0.99^2\cdot0.94\cdot0.93\approx0.8568[/tex]

Answer:

3.5 seconds

Step-by-step explanation:

h(t) is a quadratic function, it indicate that the object start with initial height (56).

If you want to know when the object hit the ground (h=0) you have to use the quadratic formula [tex](-b +- \sqrt{b^{2}-4ac } )/2a[/tex] and take the positive root (the negative shows a negative time, so we have to discard it).

In this case: a=-6, b=5 and c=56, then the solve is 7/2=3.5

Answer:

  5 meters

Step-by-step explanation:

The height of the lamppost above the person is twice the height of the person, so the distance between the lamppost and person is twice the length of the person's shadow. (A diagram can help you see this.)

The person's shadow is (10 m)/2 = 5 m long.

___

Check

The tip of the shadow is 15 m from the lamppost, 2.5 times the height of the lamp. The tip of the shadow is also 5 m from the person, 2.5 times the height of the person. The triangles involved are similar.

Final answer:

To find the length of the person's shadow, we use the properties of similar triangles defined by the person and the lamppost. By setting up a proportion between the person's and the lamppost's height to their respective shadow lengths and solving, we find the person's shadow is 5 meters long.

Explanation:

To solve the problem of determining the length of the shadow cast by the person standing 10 meters from the lamppost at night, we can use the concept of similar triangles.

Since the light source (lamppost) is above ground level, the triangle formed by the lamppost, the end of the shadow, and the top of the person's head is similar to the triangle formed by the person, their shadow, and the ground. Using the properties of similar triangles, the ratios of corresponding sides are equal.

Let's denote the length of the person's shadow as s. The triangles' corresponding sides' ratios would be:

Setting up the proportion, we have:

2 / s = 6 / (10 + s)

By cross-multiplying and solving for s, we get:

2(10 + s) = 6s

20 + 2s = 6s

4s = 20

s = 5

Hence, the length of the person's shadow is 5 meters.

Answer:

128 posibilities

Step-by-step explanation:

We have 7 colleges (A,B,C,...,H) which form a set with seven elements.

What you are asking is the number of elements (or cardinality) of the set that contains all possible sets formed by those 7 elements (or the "power set").

It is known that if n is the number of elements of a given set X, then the cardinality of the power set is [tex]2^n[/tex].

Therefore, there are [tex]2^7[/tex] or 128 possibilities (or elements) for the set of colleges that he applies to.

Final answer:

The number of possible combinations of colleges that Jose can apply to from 7 options is 128. This includes the possibility of not applying to any college as well.

Explanation:

The question asks how many different combinations of colleges Jose may apply to given 7 different options.

This is a problem related to the field of combinatorics in mathematics, specifically the concept of the power set, where each college can either be chosen or not, resulting in 2⁷ possible combinations.

Since he can like none, some, or all colleges, we include the possibility of an empty set, leading to a total of 2⁷ = 128 possibilities.

In each case, Jose has two options for every college - to apply (like) or not to apply (dislike).

Therefore, the number of combinations is calculated by raising 2 (the number of options for each item) to the power of 7 (the number of items).

Answer:D

THE POPULATION OF SPECIES A IS DECREASING. AND IT HAD THE SMALLER INITIAL POPULATION

The statement that correctly compares the given functions is - 'The population of species A is decreasing, and it had the smaller initial population.'

The correct answer is an option (D)

"A function of the form [tex]f(x)=b^x[/tex] where b is constant."

" [tex]f(x) = a (1 + r)^x[/tex]

where a is the initial value

r is the growth rate

x is time"

For given question,

We have been given a exponential function [tex]f(x) = 1400(0.70)^x[/tex]

This function represents the population of species A, x years from today.

The population of species B is modeled by function g, which has an initial value of 1,600 and increases by 20% per year.

a = 1600

r = 20%

 = 0.2  

Using the exponential growth formula the exponential function that represents the population of species B would be,

[tex]g(x) = 1600 (1 + 0.2)^x\\\\g(x)=1600(1.02)^x[/tex]

We know that, if the factor b ([tex]f(x)=a\bold{b}^x[/tex]) is greater than 1 then the exponential function represents the growth and if b < 1 then the exponential function represents the decay of population.

From functions f(x) and g(x) we can observe that, the population of species A is decreasing, and it had the smaller initial population.

So, the correct answer is an option (D)

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Answer:

Se below.

Step-by-step explanation:

The Chord Intersection Theorem:

If 2 chords of a circle are AB and CD and they intersect at E, then

AE * EB = CE * ED.

Problem.

Two Chords AB and CD intersect  at E.  If AE =  2cm , EB = 4 and CE = 2.5 cm, find the length of ED.

By the above theorem : 2 * 4 = 2.5 * ED

ED = (2 * 4) / 2.5

The measure of ∠XYZ is 35°.

Secants theorem states that the angle formed by the two secants which intersect inside the circle is half the sum of the intercepted arcs.

Here is the problem of chords that we would use the secants theorem

Circle A is shown. Secant W Y intersects tangent Z Y at point Y outside of the circle. Secant W Y intersects circle A at point X. Arc X Z is 110° degrees and arc W Z is 180° degrees. In the diagram of circle A, what is the measure of ∠XYZ?

We want to determine the angle ∠XYZ in the image attached.

To solve that, we will use the formula in the theorem for angles formed by secants or tangents. Thus;

According to Secants theorem,

∠XYZ = ½(arc WZ - arc XZ)

Given, arc WZ = 180° and arc XZ = 110°

Thus;

∠XYZ = ½(180 - 110)

∠XYZ = ½(70)

∠XYZ = 35°

Hence, the measure of ∠XYZ is 35°.

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Answer:

The answer to your question is: h = 9 in

Step-by-step explanation:

Data

radius = 6 inches

V = 324 π in³

h = ?

Formula

V = πr²h

Then, we solve it

324π = πr²h          cancel π

 324 = r²h             substitution

324 = 6² h

h = 324 /6²           simplify

h = 324/36

h = 9 in              result

Utilizing the volume formula for a cylinder, we can calculate the minimum height required for a volume of 324π cubic inches. Given a radius of 6 inches, we solve the formula to find a minimum cylinder height of 9 inches.

The subject of this question is mathematics, specifically geometry focusing on the properties of cylinders. In this instance, we want to calculate the minimum height of a cylinder that can hold a specified volume. The formula to find the volume of a cylinder is V = πR²h, where 'R' is the radius and 'h' is the height.

Given that the radius is 6 inches and the desired volume is 324π cubic inches, we can insert these values into our formula. So, 324π = π*(6)²*h which simplifies to 324 = 36h. Then, we need to solve for 'h', which involves dividing 324 by 36. The result shows that the minimum height 'h' of the cylinder is 9 inches.

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Answer:

[tex]y=58.67^{\circ}[/tex]

[tex]x=121.33^{\circ}[/tex]

Step-by-step explanation:

We are given that two angles are supplementary .

We have to write an equation that can be used to find the measures.

Let x and y are supplementary

According to question

[tex]x+y=180^{\circ}[/tex]( by definition of supplementary angles)

[tex]x=2y+4[/tex]

Substitute the value then we get

[tex]2y+4+y=180[/tex]

[tex]3y=180-4[/tex]

[tex]3y=176[/tex]

[tex]y=\frac{176}{3}[/tex]

[tex]y=58.67^{\circ}[/tex]

Substitute the value then, we get

[tex]x+58.67=180[/tex]

[tex]x=180-58.67[/tex]

[tex]x=121.33^{\circ}[/tex]

Bobby's economic profit for the day is $0.60, calculated by subtracting his total cost of $3.40 from his total revenue of $4.00. Here option A is correct.

To calculate Bobby's economic profits, we first need to understand the concept of economic profit.

Economic profit is calculated as total revenue minus total cost. Total revenue (TR) is the total amount of money earned from selling a product, which is calculated by multiplying the quantity sold (Q) by the price per unit (P). Total cost (TC) is the total expense incurred in producing a product.

In this case, we have the following information:

Bobby sells 20 glasses of lemonade at $0.20 per cup, so his total revenue is:

TR = 20 cups * $0.20/cup = $4.00

Bobby's average total cost is $0.17 per cup. Since he sold 20 cups, his total cost is:

TC = 20 cups * $0.17/cup = $3.40

Now, we can calculate Bobby's economic profit:

Economic Profit (π) = Total Revenue (TR) - Total Cost (TC)

= $4.00 - $3.40

= $0.60

So, Bobby's economic profit for the day is $0.60.

The correct option is:

a. $0.60

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The coordinates of point Q are (0, 3).

Option C is the correct answer.

The coordinates in a graph indicate the location of a point with respect to the x-axis and y-axis.

The coordinates in a graph show the relationship between the information plotted on the given x-axis and y-axis.

We have,

From the graph,

Point Q is at y = 3 on the y-axis and 0 on the x-axis.

So,

The coordinates of point Q are (0, 3).

Thus,

The coordinates of point Q are (0, 3).

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American Airlines used cluster sampling by selecting entire flights (clusters) and surveying every passenger on those flights.

To determine customer opinion of their check-in service, American Airlines employs a specific type of sampling method by randomly selecting 70 flights during a certain week and surveying all passengers on those flights. This is an example of cluster sampling, which is one of the probability sampling techniques.

In cluster sampling, the population is divided into clusters (e.g., flights in this case) and then entire clusters are randomly selected. All individuals within the chosen clusters are included in the sample. The key element here is that entire clusters are selected, and every member of those clusters is surveyed.

The maximum volume of such a box is determined by forming an equation that represents the condition about the width of the box and the perimeter of one of its nonsquare sides, then finding the maximum value of the volume function obtained by differentiating the function and setting it equal to zero.

The subject of this question is mathematics related to box dimensions, more specifically the geometry dealing with volume of a box. The box in question has a square base, which means the length and width are the same, let's call this x. Because the box is wider than it is tall, we know it's height, let's call it h, is less than x.

According to the question, the width of the box and the perimeter of one of the nonsquare sides of the box sum to no more than 117 inches. The perimeter of a nonsquare side (a rectangle) is given by 2(x+h), and if we add x (the width of the box) to this, we get x + 2(x+h) which must be less than or equal to 117. Simplifying gives 3x + 2h <= 117

We are interested in the volume of the box which can be determined by multiplying the length, width, and height (V = x*x*h). This can be simplified to V = x^2 * h. To get the maximum volume, we should make h as large as possible. Substituting 3x into the original inequality for h (since 3x <= 117), we get h <= 117 - 3x. Thus, the volume V becomes V = x^2 * (117 - 3x).

To find the maximum volume, we take the derivative of the volume function (V = x^2 * (117 - 3x)) with respect to x and set it equal to zero. This will give us the value of x for which the volume is maximum. Once we have the value of x, we substitute it back into the volume function to get the maximum volume.

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The maximum volume for such a box is 152,882.5 cubic inches

We have a box with a square base, where the width w is greater than the height h. The constraint given is that the width of the box plus the perimeter of one of the non-square sides cannot exceed 117 inches.

The perimeter of one of the non-square sides is 2h + 2d, where d is the depth. Therefore, the constraint equation becomes:

[tex]\[ w + 2h + 2d \leq 117 \][/tex]

Since the base is square, w = h. Let's denote this common value as s. So, the constraint equation simplifies to:

[tex]\[ 2s + 2d \leq 117 \][/tex]

Now, we need to express the volume of the box in terms of s and d:

[tex]\[ V = s^2d \][/tex]

We want to maximize V subject to the constraint [tex]\(2s + 2d \leq 117\).[/tex]

To proceed, let's solve the constraint equation for d:

[tex]\[ d \leq \frac{117 - 2s}{2} \][/tex]

Since d must be greater than zero, we have:

[tex]\[ 0 < d \leq \frac{117 - 2s}{2} \][/tex]

Now, we substitute [tex]\(d = \frac{117 - 2s}{2}\)[/tex] into the volume equation:

[tex]\[ V(s) = s^2 \left( \frac{117 - 2s}{2} \right) \]\[ V(s) = \frac{1}{2}s^2(117 - 2s) \][/tex]

To find the maximum volume, we'll take the derivative of V(s) with respect to s, set it equal to zero to find critical points, and then check for maximum points.

[tex]\[ V'(s) = \frac{1}{2}(234s - 6s^2) \]\\Setting \(V'(s)\) equal to zero:\[ \frac{1}{2}(234s - 6s^2) = 0 \]\[ 234s - 6s^2 = 0 \]\[ 6s(39 - s) = 0 \][/tex]

This gives us two critical points: s = 0 and s = 39. Since s represents the width of the box, we discard s = 0 as it doesn't make physical sense.

Now, we need to test s = 39 to see if it corresponds to a maximum or minimum. Since the second derivative is negative, s = 39 corresponds to a maximum.

So, the maximum volume occurs when s = 39 inches.

Substitute s = 39 into the constraint equation to find d:

[tex]\[ 2(39) + 2d = 117 \]\[ 78 + 2d = 117 \]\[ 2d = 117 - 78 \]\[ 2d = 39 \]\[ d = \frac{39}{2} \]\[ d = 19.5 \][/tex]

Therefore, the maximum volume of the box is achieved when the width and height are both 39 inches, and the depth is 19.5 inches. Let's calculate the maximum volume:

[tex]\[ V_{\text{max}} = (39)^2 \times 19.5 \]\[ V_{\text{max}} = 152,882.5 \, \text{cubic inches} \][/tex]

So, the maximum volume for such a box is 152,882.5 cubic inches.

Answer:

15

Step-by-step explanation:

Using the definition of n[tex]C_{r}[/tex]

n[tex]C_{r}[/tex] = [tex]\frac{n!}{r!(n-r)!}[/tex]

where n! = n(n - 1)(n - 2) × 3 × 2 × 1

Hence

6[tex]C_{2}[/tex]

= [tex]\frac{6!}{2!(4!)}[/tex]

= [tex]\frac{6(5)(4)(3)(2)(1)}{2(1)4(3)(2)(1)}[/tex]

Cancel 4(3)(2)(1) on numerator/ denominator, leaving

= [tex]\frac{6(5)}{2(1)}[/tex] = [tex]\frac{30}{2}[/tex] = 15