Area and Volume Question 4

calculator allowed A builder has drawn a trapezoid in the $xy$-plane above to represent the stage at a concert. If each unit of the coordinate axes represents 5 feet, what is the area, in square feet, of the stage?

$ 32$

$40$

$ 800 $

$ 1{,}000$

Approach 1

The area of a trapezoid is given by the following formula:

$$ A=\frac{1}{2}(b_1+b_2)h$$

A nice way to remember the area formula of a trapezoid is to think of it as multiplying the average of the bases with the height.

$$ A=\underbrace{\frac{1}{2}(b_1+b_2)}_\text{average of bases}h$$

Before using our equation, we should convert each unit to its length in feet. $ \\ b_1=6 \text{ units} = 30 \text{ feet} \\ $ $ b_2=10 \text{ units} = 50 \text{ feet} \\ $ $ h=4 \text{ units} = 20 \text{ feet} $ $$ A = \frac{1}{2}(b_1+b_2)h $$ $$ A = \frac{1}{2}(30+50)20 $$ $$ A = \boxed{800 \text{ ft}^2} $$

Approach 2

We can take a similar approach but use the area formulas for triangles and rectangles, which are easier to remember than the formula for a trapezoid.

We can find the total area by adding the areas of the two equivalent triangles with the area of the rectangle. Before using our equations, we should convert each unit to its length in feet.

For the triangle, $ \\ b=4 \text{ units} = 20 \text{ feet} \\ $ $ h=2 \text{ units} = 10 \text{ feet} \\ $

$$A_{\text{triangle}}= \frac{1}{2}bh$$ $$A_{\text{triangle}}= \frac{1}{2}(10)20$$ $$A_{\text{triangle}}= 100 \text{ ft}^2 $$

For the rectangle, $\\ l=6 \text{ units} = 30 \text{ feet} \\ $ $ w=4 \text{ units} = 20 \text{ feet} $

$$ A_{\text{rectangle}}= lw $$ $$A_{\text{rectangle}}= 30(20) $$ $$A_{\text{rectangle}}= 600 \text{ ft}^2 $$

$$ A_{\text{trapezoid}}=2A_{\text{triangle}}+A_{\text{rectangle}} $$ $$ A_{\text{trapezoid}}=2(100)+600 $$ $$ A_{\text{trapezoid}}=\boxed{800 \text{ ft}^2} $$