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dlinn [17]
3 years ago
5

A rectangle is inscribed under the curve y = 24 – x2 with a portion of the x-axis as its base. What is the area, in square units

, of the largest such rectangle?
Mathematics
1 answer:
8090 [49]3 years ago
3 0
Answer: 90.51 square units

Explanation:

1) The rectangle has one length on the x-axis.

2) Call x, the x-coordinate of the right lower corner of the rectangle.

And the point is (x,0)

3) The x-coordinate of the lef lower corner will be - x.

And the point is (-x,0)

4) The right upper point of the rectangle will be (x,y) where y = 24 - x²

5) The left upper point of the rectangle will be (-x,y) where y = 24 - x² .

6) The area of such rectantle is the length of the base times the length of the height.

The length of the base is 2x

The lenght of the height is y

So, the area is A = 2x(y) = 2x(24 - x²) = 48x - 2x³

7) Now to find the maximum area you have to derivate the function and make it equal to zero:

dA
---- = A' = 48 - 6x² = 0 =>
dx

4x² = 48 => x² = 48/6 = 8 =>

x = (+/-)√8

Then, y = 24 - x² = 24 - (√8)² = 24 - 8 = 16

8) Therefore the area of the rectangle is

A = 2x(y) = 2(√8)(16) = 90.51.

9) You can see if x is a greater or smaller.

For example x = 2 and x = 3

x = 2 => y = 24 - (2)² = 20 => A = 2(2)(20) = 80 which is less than 90.51

x = 3 => y = 24 - (3)² = 15 => A = 2(3)(15) = 90 which is less than 90.51

So, that tells you that your result should be right.
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The tip is not included in the tax, and the tax does not include the tip. You can shorten this up quite a bit.
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There's a problem. He can't buy 0.72 of a Latte. He can only buy 10 of them. In this case we must round down so what happens if he buys 10 lattes? Does the price the taxes and the tip allow him to get another Latte? The answer should be no because then you would be 11 lattes and the tip and taxes would push you over 60 dollars. But be careful, sometimes things like this can alter you answer.. 

Answer: 10 <<<<
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