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805+ (Hard)|   Geometry|      
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Bunuel
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The figure shows a window that is in the shape of a rectangle with 16 Nov 2022, 11:48
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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95% (hard)

Question Stats:

33% (03:18) correct 67% (02:29) wrong based on 24 sessions

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The figure shows a window that is in the shape of a rectangle with a semicircle at the top. The radius of the semicircle is x feet, and the perimeter of the window is 26 feet. What is the total area, in square feet, of the window in terms of x?

A. \(13x-(π+2)x^2\)

B. \(26x-2x^2\)

C. \(26x-(\frac{π}{2}+2)x^2\)

D. \(26x-(\frac{π}{2}+4)x^2\)

E. \(26x-(\frac{3π}{2}+2)x^2\)

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C
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divyadna
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The figure shows a window that is in the shape of a rectangle with Updated on: 17 Nov 2022, 02:30
Originally posted by divyadna on 16 Nov 2022, 12:43.
Last edited by divyadna on 17 Nov 2022, 02:30, edited 2 times in total.
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The figure shows a window that is in the shape of a rectangle with 16 Nov 2022, 18:09
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You made a mistake, you’re including “2x” twice when it should only be included once. After doing this, you should get option C, which is the correct choice.

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dsp9846
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The figure shows a window that is in the shape of a rectangle with 16 Nov 2022, 20:36
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Let's say the length of the window is y,

The radius of the semicircle is x that is given.

now based on that perimeter of the window = 2x+2y+(2πx)/2 = 2x + 2y + πx

now perimeter is given 26 feet,

so, 2x + 2y + πx = 26

2y = 26 - 2x - πx

Now the area of the window = area of the rectangle portion + area of the semicircle
= (y)(2x) + (πx^2)/2
= (x)(2y) + (πx^2)/2
= (x)(26 - 2x - πx) + (πx^2)/2
= 26x - 2x^2 - πx^2 + (πx^2)/2
= 26x - 2x^2 - (πx^2)/2
= 26x−(π/2+2)x^2

So answer is option C
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The figure shows a window that is in the shape of a rectangle with 17 Nov 2022, 04:47
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Bunuel

The figure shows a window that is in the shape of a rectangle with a semicircle at the top. The radius of the semicircle is x feet, and the perimeter of the window is 26 feet. What is the total area, in square feet, of the window in terms of x?

A. \(13x-(π+2)x^2\)

B. \(26x-2x^2\)

C. \(26x-(\frac{π}{2}+2)x^2\)

D. \(26x-(\frac{π}{2}+4)x^2\)

E. \(26x-(\frac{3π}{2}+2)x^2\)

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Solution:

  • Let us assume the length of the rectangle be \(l\)
  • Perimeter of the window = perimeter of the dome + 3 sides of rectangle
    \(⇒26=\pi x+l+l+2x\)
    \(⇒26=\pi x+2l+2x\)
    \(⇒2l=26-\pi x-2x\)
    \(⇒l=13-\frac{\pi x}{2}-x\)
  • Area of the window = area of dome + area of rectangle
    \(=\frac{\pi x^2}{2}+2lx\)
    \(=\frac{\pi x^2}{2}+2(13-\frac{\pi x}{2}-x)x\)
    \(=\frac{\pi x^2}{2}+26-\pi x^2-2x^2\)
    \(=26-\frac{\pi x^2}{2}-2x^2\)
    \(=26-x^2(\frac{\pi}{2}+2)\)

Hence the right answer is Option C
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