Without changing the front and back widths, by how much must Boris increase the depth of his garden to triple the area?

Question

Boris has a rock garden in the shape of a trapezoid. The garden has an area of 90 m² in a depth of 12 m. The front width is four times the length of the back width. Without  changing the front and back width,  by how much must force increase the depth of his garden to triple the area.

Answer 1

A=(1/2)(h)(x+4x) where x and 4x are the parallel bases

90=(1/2)(12)(5x)

90=(6)(5x)

90=30x

x=3 and 4x=12

90=(1/2)(12)(3+12)

90=(1/2)(12)(15)

if you want to triple the area without changing the bases, you must change the depth

270=(1/2)(h)(15)

540=(h)(15)

540/15=h

h=36

you must triple the depth so you must increase the depth by 24 m to get the depth equal to 36 m

Answer 2

If the front and back widths must remain a constant, then to triple the area, you must triple the depth. In this case, since the original depth is 12 meters, then Boris would need to have a depth of 36 meters to increase his area from 90 square meters to 270 square meters.

 

BTW...The original back width is 3 meters while the original front width is 12 meters. One does not need to know this in order to solve this problem, but I thought I would mention it anyway. The key point is that the area is calculated by knowing the sum of the two widths and the depth. These are multiplied together and divided by 2 to determine the area.

 

The exact formula is A = 0.5 * d * (w1 + w2)    where d is the depth and w1 and w2 are the front and back widths.

 

This formula is usually presented as A = 0.5 * h * (b1 + b2) where h is the height and b1 and b2 are the bases (the two parallel sides of the trapezoid.