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705-805 (Hard)|   Geometry|      
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Bunuel
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An interior designer decides to accent a wall with an evenly spaced 28 Aug 2023, 01:30
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85% (hard)

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46% (02:42) correct 54% (03:40) wrong based on 35 sessions

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An interior designer decides to accent a wall with an evenly spaced row of stenciled circles. The wall is 31 feet 6 inches long, and each stenciled circle has an area of 36π square inches. If the designer wants to leave a space of x inches between each circle and at either end of the row, with no space left over, and x must be an integer, then what is the greatest possible number of circles that the designer can use?

A. 28
B. 29
C. 30
D. 31
E. 32



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B
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An interior designer decides to accent a wall with an evenly spaced 28 Aug 2023, 07:12
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Bunuel
An interior designer decides to accent a wall with an evenly spaced row of stenciled circles. The wall is 31 feet 6 inches long, and each stenciled circle has an area of 36π square inches. If the designer wants to leave a space of x inches between each circle and at either end of the row, with no space left over, and x must be an integer, then what is the greatest possible number of circles that the designer can use?

A. 28
B. 29
C. 30
D. 31
E. 32



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Area (circle) =36π square inches
πr^2 = 36
radius = 6
diameter = 12

1ft = 12 inches
Total wall = 378 inches (ie 372 + 6 inches)

Given: (12+x) x number of circle = 378
In order to max the no of circles, we will minimize x
therefore x = 1
13 x no of circles = 378
no of circles = approx 29
Hence 29
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An interior designer decides to accent a wall with an evenly spaced 03 Sep 2023, 10:17
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Bunuel
An interior designer decides to accent a wall with an evenly spaced row of stenciled circles. The wall is 31 feet 6 inches long, and each stenciled circle has an area of 36π square inches. If the designer wants to leave a space of x inches between each circle and at either end of the row, with no space left over, and x must be an integer, then what is the greatest possible number of circles that the designer can use?

A. 28
B. 29
C. 30
D. 31
E. 32





Area (circle) =36π square inches
πr^2 = 36
radius = 6
diameter = 12

1ft = 12 inches
Total wall = 378 inches (ie 372 + 6 inches)

Given: (12+x) x number of circle = 378
In order to max the no of circles, we will minimize x
therefore x = 1
13 x no of circles = 378
no of circles = approx 29
Hence 29
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Could you explain why it is 12+x? I'm thinking, it should be 12+2x(number of circles) + x, since there are 2 x's at either end of the row. so the last x counts as the x in the last 12+x of the last circle. The additional x then will be the x at the front
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An interior designer decides to accent a wall with an evenly spaced 04 Sep 2023, 08:22
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Bunuel
An interior designer decides to accent a wall with an evenly spaced row of stenciled circles. The wall is 31 feet 6 inches long, and each stenciled circle has an area of 36π square inches. If the designer wants to leave a space of x inches between each circle and at either end of the row, with no space left over, and x must be an integer, then what is the greatest possible number of circles that the designer can use?
A. 28
B. 29
C. 30
D. 31
E. 32
Show more

Solution:
  • Let the total number of circles be \(n\)
  • To maximise the number of circles i.e., \(n\), we have to minimise the space i.e., \(x\)
  • Minimum space will be 1 inch. So, lets assume \(x=1\)
  • And number of space (between circles and circle and wall edge) will be one more than the number of circles.
  • So, number of spaces of 1 inch each \(=n_{max}+1\)

  • Area of circle \(=36\pi\). So, radius \(=6\) and diameter \(=12\) inches
  • Length of wall = 31 feet 6 inches \(= 31\times 12+6=378\) inches

  • According to the problem, \(12n_{max}+n_{max}+1=378\)
    \(⇒13n_{max}=377\)
    \(⇒n_{max}=\frac{377}{13}=29\)

Hence the right answer is Option B
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