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Maru [420]
4 years ago
7

Find the area bounded by the given curves: y=2x−x2,y=2x−4

Mathematics
1 answer:
Andrej [43]4 years ago
7 0

Answer:

A = [\frac{32}{3}]

Step-by-step explanation:

Given

y_1 = 2x - x^2

y_2 = 2x - 4

Required

Determine the area bounded by the curves

First, we need to determine their points of intersection

2x - x^2 = 2x - 4

Subtract 2x from both sides

-x^2 = -4

Multiply through by -1

x^2 = 4

Take square root of both sides

x = 2   or    x = -2

This Area is then calculated as thus

A = \int\limits^a_b {[y_1 - y_2]} \, dx

<em>Where a = 2 and b = -2</em>

Substitute values for y_1 and y_2

A = \int\limits^a_b {(2x - x^2) - (2x - 4)} \, dx

Open Brackets

A = \int\limits^a_b {2x - x^2 - 2x + 4} \, dx

Collect Like Terms

A = \int\limits^a_b {2x - 2x- x^2  + 4} \, dx

A = \int\limits^a_b {- x^2  + 4} \, dx

Integrate

A = [-\frac{x^{3}}{3} +4x](2,-2)

A = [-\frac{2^{3}}{3} +4(2)] - [-\frac{-2^{3}}{3} +4(-2)]

A = [-\frac{8}{3} +8] - [-\frac{-8}{3} -8]

A = [\frac{-8+ 24}{3}] - [\frac{8}{3} -8]

A = [\frac{-8+ 24}{3}] - [\frac{8-24}{3}]

A = [\frac{16}{3}] - [\frac{-16}{3}]

A = [\frac{16}{3}] + [\frac{16}{3}]

A = [\frac{16 + 16}{3}]

A = [\frac{32}{3}]

Hence, the Area is:

A = [\frac{32}{3}]

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Answer:

<h3>(-12, 2)</h3>

Step-by-step explanation:

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\underline{+\left\{\begin{array}{ccc}-2x-2y=-16\\2x+3y=36\end{array}\right}\qquad\text{add both sides of the equations}\\.\qquad\qquad y=20\\\\\text{Put the value of y to the firast equation:}\\\\x+20=8\qquad\text{subtract 20 from both sides}\\x=-12\\\\\boxed{(-12,\ 20)}

8 0
3 years ago
What is the equation of this line in slope-intercept form?
nignag [31]

Answer:

y = -3x + 2

Step-by-step explanation:

y = mx + b

m is the slope and b is the y-intercept

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Sixth grade mathematics
Tpy6a [65]

Hello!

Answer:

1 quart = 4 cups

7 quarts = 28 cups

1 guest needs 2 cups

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Hope this helps! Have a day <3

3 0
3 years ago
In the morning, Marco sold 12 cups of lemonade for $3. By the end of the day, he had earned $9. How many cups of lemonade did he
MA_775_DIABLO [31]

Answer:

Marco earned $2.25.

Step-by-step explanation:

1. If Marco sold 12 cups of lemonade for 3$, we would divide 12 cups and how much he earned from the 12 cups (3$). 12/3 = 4

2. By the end of the day, Marco earned $9. So, 4 cups = 1$. To break it down, 1$ = 100 cents and 100 cents divided by 4 is 25 cents.  Cups to Cost. ( 4 : 1 ) and/or ( 1 : 0.25 )

3. 8 cups = 2$ plus 1 cup (0.25) = 2.25.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

7 0
3 years ago
Read 2 more answers
Need Help
klio [65]

The smallest amount of material needed is 54 square centimeters

<h3>How to determine the amount of material needed?</h3>

The given parameters are:

Volume = 36 cubic centimeters

Represent length with x, width with y and height with z.

So, we have

x = 3y

The volume is calculated as:

V = xyz

This gives

V = 3y²z

Substitute 36 for V

3y²z = 36

Divide by 3

y²z = 12

Make z the subject

z = 12/y²

The surface area is:

S = 2(xy + xz + yz)

This gives

S = 2(3y² + 3yz + yz)

Evaluate the like terms

S = 2(3y² + 4yz)

Expand

S = 6y² + 8yz

Substitute z = 12/y²

S = 6y² + 8y * 12/y²

This gives

S = 6y² + 96/y

Differentiate

S' = 12y -  96/y²

Set to 0

12y -  96/y² = 0

Multiply through by y²

12y³ - 96 = 0

Add 96 to both sides

12y³ = 96

Divide by 12

y³ = 8

Take the cube root of both sides

y = 2

Recall that:

x = 3y and z = 12/y²

This gives

x = 3 * 2 = 6

z = 12/2² = 3

Recall that:

S = 2(xy + xz + yz)

So, we have:

S = 2(6 * 2 + 3 * 3 + 2 * 3)

Evaluate

S = 54

Hence, the smallest amount of material needed is 54 square centimeters

Read more about surface areas at:

brainly.com/question/76387

#SPJ1

8 0
2 years ago
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