Chapter 8.2, Problem 30PPS

a.

To determine

To determine the expression of the area of trapezoidal cross section of the dam.

Answer to Problem 30PPS

  $\frac{11h^2-150h}{10}$ square feet

Explanation of Solution

Given:

  

Length of base at the bottom $=$ 2 times the height

Length of base at the top $=$ $\frac{1}{5}$ times the height minus $30$ feet

Formula used:

Area of Trapezoid $=\frac{1}{2}\times$ (Sum of parallel sides) $\times$ (Distance between them)

Calculation:

Length of base at the bottom $=$ 2 times the height.

Length of base at the top $=$ $\frac{1}{5}$ times the height minus $30$ feet.

Length of base at the bottom be $l_b.$

Length of base at the top be $l_t.$

Height be $h.$

So, length of base at the bottom $l_b=2\times h$ feet

Length of base at the top be $l_t=\left(\frac{1}{5}\times h\right)-30$ feet

  $\Rightarrow$ Area of Trapezoid $=\frac{1}{2}\times$ (Sum of parallel sides) $\times$ (Distance between them)

  $\Rightarrow$ Area of Trapezoid $=\frac{1}{2}\times \left\{\left(2\times h\right)+\left(\frac{1}{5}\times h\right)-30\right\}\times h$

  $\Rightarrow$ Area of Trapezoid $=h^2+\frac{h^2}{10}-15h$

  $\Rightarrow$ Area of Trapezoid $=\frac{11h^2-150h}{10}$ square feet

Conclusion:

Therefore, the expression of the area of trapezoidal cross section of the dam is $\frac{11h^2-150h}{10}$ square feet.

b.

To determine

To determine the area of the dam if height is $180$ feet.

Answer to Problem 30PPS

  $329400$ square feet

Explanation of Solution

Given:

  

Height is $180$ feet

Formula used:

Area of Trapezoid $=\frac{1}{2}\times$ (Sum of parallel sides) $\times$ (Distance between them)

Calculation:

Length of base at the bottom $=$ 2 times the height.

Length of base at the top $=$ $\frac{1}{5}$ times the height minus $30$ feet.

Length of base at the bottom be $l_b.$

Length of base at the top be $l_t.$

Height be $h=180$ feet

So, length of base at the bottom $l_b=2\times h$ feet

Length of base at the top be $l_t=\left(\frac{1}{5}\times h\right)-30$ feet

  $\Rightarrow$ Area of Trapezoid $=\frac{1}{2}\times$ (Sum of parallel sides) $\times$ (Distance between them)

  $\Rightarrow$ Area of Trapezoid $=\frac{1}{2}\times \left\{\left(2\times h\right)+\left(\frac{1}{5}\times h\right)-30\right\}\times h$

  $\Rightarrow$ Area of Trapezoid $=h^2+\frac{h^2}{10}-15h$

  $\Rightarrow$ Area of Trapezoid $=\frac{11h^2-150h}{10}$ square feet

Putting, $h=180$ feet

  $\Rightarrow$ Area of Trapezoid $=\frac{11{(180)}^2-150(180)}{10}$ square feet

  $\Rightarrow$ Area of Trapezoid $=\frac{329400}{10}$ square feet

  $\Rightarrow$ Area of Trapezoid $=329400$ square feet

Conclusion:

Therefore, the area of the dam if height is $180$ feet is $329400$ square feet.