A rectangle has a perimeter of 100 inches, and

one side has length x. Express the area of the

rectangle as a function of x.


Use the function in part a to find the dimensions of the rectangle with permitter 100 inches and the largest possible area

Answers

Answer 1

Answer:

a. A = 50x - x² b. length = 25 inches and width = 25 inches and the maximum area is 625 in²

Step-by-step explanation:

a. The perimeter of a rectangle P = 2(L + W) where L = length and W = width. Now, given that P = 100 inches and W = x, substituting these into the equation, we have

P = 2(L + W)

100 = 2(L + x)

dividing both sides by 2, we have

100/2 = L + x

50 = L + x

making L subject of the formula, we have

L = 50 - x

Now, the are of a rectangle A = LW. Substituting the values of L and W, we have

A = LW

A = (50 - x)x

A = 50x - x²

b. To find the largest possible area of rectangle with perimeter 100 inches, we differentiate A and equate it to zero to find the value of x that maximizes A.

So, dA/dx = d(50x - x²)/dx

dA/dx = d50x/dx - dx²/dx

dA/dx = 50 - 2x

dA/dx = 0 ⇒ 50 - 2x = 0

50 = 2x

dividing both sides by 2, we have

x = 50/2

x = 25

To find it this gives maximum value for A, we differentiate A twice.

d²A/dx² = d(50 - 2x)/dx

d²A/dx² = d50/dx - d2x/dx

d²A/dx² = -2

Since d²A/dx² = -2 < 0, so x = 25 gives maximum value for the area, A.

Since W = x = 25 in and L = 50 - x. So, L = 50 - 25 = 25 in

So, the maximum area A = LW = Lx = 25 in × 25 in = 625 in²

The dimension with perimeter 100 inches that give maximum area are length = 25 inches and width = 25 inches and the maximum area is 625 in²


Related Questions

b: 6 Point: (-4, 2)
Solve for y=mx+b
Please help

Answers

Answer:

y=-4x+2

Step-by-step explanation:

B is the y-intercept and M is the sope, or x, knowing this now we slove:

(-4, 2)

m    b

y=mx+b

y=-4x+2

Have a beautiful day! v(⌒o⌒)v♪

For more help take a look at the picture below.

Will give brainliest.

Answers

Answer:

y = -2/3x +3

Step-by-step explanation:

Determine A:B:C, if A:B= 2:3 and B:C =4: 5​

Answers

I’m confused what we are suppose to do

Find the common difference of the arithmetic sequence 13, 18, 23, ...

Answers

Answer:

5

Step-by-step explanation:

To find the common difference you just find the difference between each number. 18-13=5,  23-18=5,  etc.

Answer: awnswer mine for 53 points

Step-by-step explanation:

slope intercept
y= 1/2x + 2

Answers

Don't know exactly what you're asking, but I think you're asking what the slope and what the y-intercept is.

1/2x is the slope

Positive 2 is the y-intercept

it y because of the equation mx +b and b is y

Twelve large bottles of water cost $9.How many bottles can you buy for $3?

Answers

Answer: 4 large bottles of water for $3.

Step-by-step explanation: To find how many large water bottles you can buy with $3, you’ll have to divide 12 by 3 which equals to 4. You can check your answer by multiplying 4 x 3 which equals to 12, which is the original amount of large water bottles you can buy for $9.

Bob went out to dinner and paid a 15% tip on a $20 meal? How much is the total bill?

Answers

Answer:

23

Step-by-step explanation:

15% of 20 = 3  then add the 3 to the 20 to get 23

23Answer:

Step-by-step explanation:

15/100*20 = 3

20+3 = 23

Darryl wants to improve the food at his restaurant, Downtown Deli, by purchasing some new items to sell. He wants to save as much money as possible. First, Darryl is pricing brands of gourmet pickles. What is the unit price for each brand of pickle? Brand A: Brand B: Brand C: Brand D:

Answers

Answer:

Brand A: $0.40

Brand B:  $0.30

Brand C: $0.25

Brand D: $0.20

Answer:

Step-by-step explanation:

1. 0.40

2. 0.30

3. 0.25

4. 0.20

Which graph below correctly shows the two lines on the same axes ?

Answers

Answer:

The one in the top left corner I THINK

Step-by-step explanation:

It is on baisically the same line and makes the most sense to ME.

DO NOT PICK ME IF YOU ARE NOT SURE!!!

Factoring Polynomials Completely

Answers

The answer would be

(x+2)

The first option

Factoring [tex]6x^2+7x-10[/tex]

gives us:

[tex]\left(6x-5\right)\left(x+2\right)[/tex]

so we know that the first option,

one of the factors is (x+2) is correct.

Help me please lol!!!!!!

Answers

Answer:

y<4

Step-by-step explanation:

thats your answer :)

5×(-5+6) distributive law​

Answers

Answer:

5×(+1)=

5×1=5

Step-by-step explanation:

hope it helps

Answer:

Step-by-step explanation:

if you're going for distributive law then 5(-5)+5(6). -25+30. so 5

Which expression is equal to 5 6 (6 is an exponent).

6x5

36x6x6x6

25x5x5x5

125x125

Answers

Answer: 125x125

Step-by-step explanation:

125 x 125 is the correct answer. Both equal 15,625.

an elevator starts at the 6th floor it goes up 7/4 plus and then it goes down 10 floors. a second elevator starts on the 20th floor. it goes down five floors three times, and then it goes up 12 floors. which elevator ends up on a higher floor?​

Answers

Answer:

Elevator 2

Step-by-step explanation:

Well let's make this simply by focusing on elevator number 1

it is on the 6th floor so we represent that as

6

Now it goes up 7/4 which is also the same as 1 3/4

the equation looks like this:

6+1 3/4

Add them together:

7 3/4

Now it goes down 10 meaning it is -10 add that to the equation:

7 3/4 - 10

Which is -2 1/4

Elevator number 1 is at floor number -2 1/4

Now for Elevator number 2

It is on the 20th floor which will make the equation looks like this:

20

Now it goes down five floors three times which can be represented like this:

20-5(3)

Simplify (Multiplication goes first according to the Order of Operations)

20-15= 5

Now it is 5 but it goes up 12 floors so the equation looks like this:

5+12

Simplify:

17

Now you could tell that elevator 2 ends up on a higher floor

Hope this helps!

Solve and graph the following solutions to
the following inequality. SHOW ALL WORK.
3
- r < -15.

Answers

Answer:

1.she is not working and the iran.

Lines j and k are perpendicular. The slope of line j is given. Find the slope of line k. m = 4/5

Answers

Answer:

A

Step-by-step explanation:

Given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{4}{5} }[/tex] = - [tex]\frac{5}{4}[/tex] → A

Bruh can someone answer this I’ll give brainliest

Answers

It’s a ! you add both of the numbers to get your missing perimeter.
I’m very sorry if it’s wrong but this is what I believe! stay safe

What is the slope of y = 4x? *

Answers

Please Brainliest me!

The slope of y=4x is 0,4 PLEASE BRAIN ME!

Solve for m. -5m=-35 m=

Answers

Answer:

m = 0

Step-by-step explanation:

Solve: Add -35m to -5m

-5m = -35m

+35m

30m = 0 *note: its okay to have 0 in this spot*

divide 0/30

m=0 because you can't divide by 0

The value of m is 7 for the given statement -5m = -35.

What are mathematical operations?

The term "operation" in mathematics refers to the process of computing a value utilizing operands and a math operator. For the specified operands or integers, the math operator's symbol has predetermined rules that must be followed. In mathematics, there are five basic operations: addition, subtraction, multiplication, division, and modular forms.

Given an expression

-5m = -35

Divide by -5 on both sides

m = 7

Therefore, For the given expression -5m = -35 the value of m is 7.

Learn more about Mathematical operations here:

https://brainly.com/question/20628271

#SPJ2

Need help on 7-8, thank you in advance! :)

Answers

Answer:

7)

a. y = 1/3x - 2

b. m = 1/3, b = -2

8.

a. y = -1/3x -5

b. m = -1/3, b = -5

Step-by-step explanation:

7.

2x - 6y = 12

-6y = -2x + 12

y = 1/3x - 2

8.

x + 3y = -15

3y = -x - 15

y = -1/3x -5

c.

Please help me i don't understand. I will give points for explanation too.

Answers

Answer:

what is the question and where is it?

Step-by-step explanation:

Where is the question, I’d be more then happy to help

PLEASE!! For 22 points!!

Write the equation of the line that passes through the points (-4,-4) and (6,3).
Put your answer in fully reduced point-slope form, unless it is a vertical or horizontal
line.

Answers

Answer:

y = 3/5x + 3/5.

Step-by-step explanation:

the /'s are division symbols.              3            3

                                                  y =     _   +       _

                                                           5            5

Write the index for 2^2 x2^3

Answers

Answer:

2^5

Step-by-step explanation:

PlS HELP QUICKLY :One of the roots of the equation x2−bx+c=0 is equal to 5. Find c in terms of b.:
Guys pls help and find what C is equal to.

Answers

Answer:First suppose that the roots of the equation

x2−bx+c=0(1)

are real and positive. From the quadratic formula, we see that the roots of (1) are of the form

b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2.

For the root or roots to be real, we require that b2−4c≥0, that is, b2≥4c. In order for them to be positive, we require that

b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0.

This immediately tells us that b>0, but we can go further. We can rearrange this to get

b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,

which (assuming that b>0) is true if and only if

b2>b2−4c,

since both sides of the inequality are positive so we may square. But then

4c>0.

That is, if the roots are real and positive then b>0 and b2≥4c>0.

Now suppose that b>0 and b2≥4c>0.

Then the roots of (1) are real since b2−4c≥0, and b>0 guarantees that the root b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√ is positive.

So it remains to show that b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0. We have that

4c>0,

so that

b2>b2−4c,

then square rooting shows that

b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,

so the roots of (1) are real and positive, as required.

Approach 2

the curve y equals x squared minus b x plus c showing two positive roots for y equals zero

This is intended to be a proof without words! We have from the diagram that:

If c, b and b2−4c are all positive, there are two real positive roots for x2−bx+c=0 (if b2=4c, we have two real positive equal roots).

If there are two real positive roots for x2−bx+c=0, then c and b are positive and b2−4c is non-negative.

(Why is the distance between the roots b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√?)

Approach 3

When solving problems about the roots of polynomials, it is often useful to find expressions those roots must satisfy and see if this tells us anything new. If α and β denote the roots of the equation, then

x2−bx+c=(x−α)(x−β)=x2−(α+β)x+αβ

and so α+β=b and αβ=c.

We also know that the roots of a quadratic equation are real if and only if the discriminant is non-negative, that is, if and only if b2−4c≥0.

Using these facts, if α and β are both real and positive, then b=α+β>0, c=αβ>0 and b2≥4c, as above.

Conversely, if b>0 and b2≥4c>0, then we know the discriminant is positive and hence both roots are real. We also have that

αβ>0(2)

and

α+β>0.(3)

As α and β are both real, by (2), we know that α and β are either both positive or both negative. However, if α and β were both negative, then (3) could not possibly hold. Hence α and β are both positive, as required.

We now sketch on a graph the region where b>0, c>0 and b2≥4c:

The curve b squared = 4 c as a quadratic with the c-axis vertical and the b-axis horizontal. The region below it is shaded.

The region of the b-c plane for which b>0, c>0 and b2≥4c

Sketch the region of the b-c plane in which the roots of the equation are real and less than 1 in magnitude.

We know that in order for the roots to be real we need b2≥4c as in the first part. We now need to find the region where

−1<b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.(4)

We have b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2≤b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2 so we only need to consider the values for which both

−1<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2andb+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.

Firstly, we will consider the values for which

b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2.

Rearranging gives

b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2−b.

So b<2, as the square root is non-negative, and we can square both sides to get

b2−4c<4−4b+b2,

which we may rearrange to find c>b−1.

We will now consider the values for which

−2<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√.

Similarly, we can rearrange to get

b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<b+2.

So b>−2, and we can, as before, square to get

b2−4c<b2+4b+4,

and hence c>−b−1.

To sketch the graph, we start by considering the boundary curves b2=4c, c=b−1 and c=−b−1, and the points at which they intersect. We can see that the two lines only intersect when b=0 and c=−1, and the lines intersect the curve when

b2=4b−4andb2=−4b−4

which rearrange to

(b−2)2=0and(b+2)2=0.

This tells us that each line intersects with the curve in only one place and so these lines must be tangent.

Is there a way we could have deduced this directly from (4)?

What do the lines being tangent signify in terms of our equation x2−bx+c=0?

Is this a representation of a well-known property of these equations?

Sketching the graph then yields the following picture:

The graph with the previous curve and the lines 4 c = 4 b minus 4 and 4 c = minus 4 b - 4. Each line touches the curve once and the two lines intersect at (0, minus 1). The region between the three lines/curves is shaded.

The shaded region is where b2≥4c, c>b−1, c>−b−1 and −2<b<2

We might notice that when we derived the inequalities c>b−1 and c>−b−1, a lot of the work we did was very similar.

We might also notice that the graph above is symmetric about the c axis. Why is this?

Does the graph give us any ideas about how we might deduce one inequality from the other?

Step-by-step explanation:

Where can i buy ps5

Answers

Answer: You can go online and type ps5 and it will get shipped to you. Or you can check Target, Bestbuy, Gamestop and etc.

Step-by-step explanation:

Answer:

gamestop or online!!

Step-by-step explanation:

hope this helps!!

-3 x (10i+6)
Your answer should be a complex number in the form a + bi where a and b are real numbers.

Answers

The Answer is:

−30xi−18x, I really don't know what you are looking for but this is what i came up with. Hope this is helpful!

In a Normal model, what percent of data is between -1 and +1 standard deviations of the mean?

Answers

Answer:

It's 99. 7% 95% and 68% for std dev respectively

Give brainliest thanks

find the intercepts or the line that corresponds to the table of values.

I have a photo to go with it :)​

Answers

X-intercept is (-3,0)
Y-intercept is (0,6)

The equation is y=2x+6
Replace y with 0 when solving for x-intercept and replace x with 0 when solving for y-intercept.

(3а-4)-4
how do you simplify?

Answers

Answer:

3a-8

Step-by-step explanation:

(3a-4) -4

= 3a-4-4

= 3a-8

3a-8 is the correct answer !!!!

IL GIVE BRAINLIST!! PLEASE HELP! THERES 2 ANSWERS JUST SAYING!!​

Answers

Answer: Option 2

Pls mark me brainilest!

Step-by-step explanation:

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