The formula for the volume of a trapezoidal prism is
Width #xx# Length #xx# Height
Subbing in the numbers for Width and Length is easy, but the Height changes from the front to the back of the wading pool.
It goes from #2# feet up to an unknown #x# number of feet.
So begin by working with the easy parts
Volume #= W xx L xx H#
#286# feet³ #= 8 xx 13 xx H#
#286= 104 xx "the average" H#
Divide both sides by #104# to solve for #H#
#2.75# feet = the average Height
The complication with trapezoids is that the Height gradually changes. In this case, it goes from #2# feet up to an unknown #x# number of feet.
To address that problem, the average Height is used.
In this case, the average of the two measurements for Height is #H =2.75#
Averages are determined by division
(the sum of the two heights) ÷ #2 =# the average, which is #2.75#
#2.75 = (2 + x)/(2)#
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Solve for #x#, already defined as "the missing height of the back wall of the pool"
1) Clear the fraction by multiplying both sides by #2# and letting the denominator cancel
#5.5 = 2 + x#
2) Subtract #2# from both sides to isolate #x#
#x = 3.5# #larr# The missing Height of the back wall of the pool
Answer
The missing Height of the back wall of the pool is #3.5# feet
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Check
In this case, checking the answer takes too long to be worth your study time, but here it is:
#1)# Sub in the values into the original equation for Volume to see if the equation is still true
Volume #=# Width #xx# Length #xx# Height
#286# feet³ should equal #8xx 13 xxH#
#286# should equal #104 xx H#
#2)# Write #H# as the average of the two heights, subbing in the answer #3.5# for one of the two heights
#286# should equal #(104)/(1) xx (2 + 3.5)/(2)#
#3)# Combine like terms and multiply the fractions
#286# should equal #((104)(5.5))/(2) #
#4)# Reduce the fraction
#286# should equal #(52)(5.5) #
#5)# Multiply
#286# does equal #286#
#Check#
