Question

The cross-section $$ABCD$$ of a swimming pool is a trapezium. It's width $$AB=14\text{m}$$ depth at the shallow end is 1.5 m and at the deep end is 8 m. Find the area of the cross-section.

Answer 1

Hint: Construct the line $$DE$$ perpendicular to the line $$BC$$. Find the area of the triangle $$DCE$$ using the formula $${\displaystyle \frac{1}{2}}\times base\times height$$. Find the area of the rectangle $$DABE$$ using the formula $$Base\times height$$. Add the two to find the answer.

Complete step-by-step answer:

We will first construct the line $$DE$$ perpendicular to the line $$BC$$. $$E$$ is the intersection point of $$BC$$and $DE$, such that $BC\mathrm{\perp }DE$

We know that the area of the triangle is $${\displaystyle \frac{1}{2}}\times base\times height$$. In the triangle $$DCE$$, the area can be calculated by substituting $$DE$$ for base and $$CE$$.
$${\displaystyle \frac{1}{2}}\times CE\times DE$$
From the figure it is evident that, $$DE=AB=14\text{m}$$ and, $$CE+EB=8\text{m}$$. Also, $$AD=EB=1.5\text{m}$$
Therefore $$CE=8-1.5=6.5\text{m}$$.
Substituting the value 6.5 for $$CE$$ and 14 for $$DE$$ in the equation $${\displaystyle \frac{1}{2}}\times CE\times DE$$.
Thus, area of the triangle $$DCE$$ = $${\displaystyle \frac{1}{2}}\times 6.5\times 14=45.5{\text{m}}^3$$

Also, the area of the rectangle is given by the formula $$Base\times height$$.
Substituting the value of base with $$AB$$ and $$AD$$ for height in the formula $$Base\times height$$ to find the area of the rectangle $$ABED$$.
Area of $$ABED$$=$$AB\times AD$$.
Substituting the value 14 for $$AB$$ and 1.5 for $$AD$$, we get
Area of $$ABED$$= $$14\times 1.5=21$$
The area of the cross-section $$ABCD$$ is equal to the sum of the section $$ABED$$ and section $$DCE$$.
Area of $$ABCD$$= Area of $$ABED$$+ Area of $$DCE$$.
Area of $$ABCD$$= $45.5+21=66.5{\text{m}}^3$
Thus the area of the cross section $$ABCD$$ is $$66.5{\text{m}}^3$$.

Note: The formula of the area of the trapezium can also be used alternatively. The area of the trapezium is given by $${\displaystyle \frac{1}{2}}\times \left(sumoftheparallelsides\right)\times heightbetweentheparallelsides$$.