Chapter 11.3, Problem 17PSC

a.

To determine

To calculate:Area of the trapezoid with upper base $5\sqrt{3}$

Answer to Problem 17PSC

Area of the trapezoid is $42\sqrt{3}$ square units.

Explanation of Solution

Given:

The length of the upper base is $5\sqrt{3}$

Concept used:

Here, we are using the formula

Area of trapezoid = $\frac{1}{2}\left(a+b\right)\times h$

(here, “ a ” is upper base, “ b ” is lower base and “ h ” is height)

Calculation:

Here, we know that Sin $\theta =\frac{P}{H}$

Sin ${30}^{\circ }=\frac{P}{H}$ (here, “ P ” is perpendicular and “ H ” is hypotenuse)

  $\frac{1}{2}=\frac{P}{12}$

  $P=6$

Similarly, Tan ${30}^{\circ }=\frac{P}{B}$ (here, “ P ” is perpendicular and “ b ” is base)

  $\frac{1}{\sqrt{3}}=\frac{6}{B}$

  $B=6\sqrt{3}$

Now, Tan ${30}^{\circ }=\frac{P}{6}$

  $\frac{1}{\sqrt{3}}=\frac{P}{6}$

  $P=2\sqrt{3}$

And, $x=3\sqrt{3}$

So, lower base = $3\sqrt{3}+6\sqrt{3}=9\sqrt{3}$

Area of trapezoid = $\frac{1}{2}\left(a+b\right)\times h$

→ $\frac{1}{2}\left(9\sqrt{3}+5\sqrt{3}\right)\times 6$

→ $42\sqrt{3}$

Conclusion: Hence, area of the trapezoid is $42\sqrt{3}$ square units.

b.

To determine

To calculate:The area of the trapezoid with upper base $12$

Answer to Problem 17PSC

The area of the trapezoid with upper base $12$ is $27\sqrt{3}$

Explanation of Solution

Given:

Length of upper base is $12$

Formula used:

Here, the formula is

Area of trapezoid = $\frac{1}{2}\left(a+b\right)\times h$

(here, “ a ” is upper base, “ b ” is lower base and “ h ” is height)

Calculation:

Here,Here, we know that Sin $\theta =\frac{P}{H}$ (here, “ P ” is perpendicular and “ H ” is hypotenuse)

Sin ${60}^{\circ }=\frac{P}{6}$

  $\frac{\sqrt{3}}{2}=\frac{P}{6}$

  $P=3\sqrt{3}$

And, Tan ${60}^{\circ }=\frac{3\sqrt{3}}{B}$

  $\sqrt{3}=\frac{3\sqrt{3}}{B}$

  $B=3$

Area of trapezoid = $\frac{1}{2}\left(a+b\right)\times h$

  = $\frac{1}{2}\left(12+6\right)\times 3\sqrt{3}$

→ $27\sqrt{3}$

Conclusion: Hence, the area of the trapezoid with upper base $12$ is $27\sqrt{3}$