Chapter 12.2, Problem 35PPS

To determine

To find: The surface area of the given composite solid.

Answer to Problem 35PPS

The surface area of the given composite solid $=2824.8{\text{cm}}^2$ .

Explanation of Solution

Given Information:

Dimensions of the given solid is shown in the given diagram

  

Formula used:

Area of trapezoid:

A trapezium is a quadrilateral in which one pair of sides are parallel, but the other pair of opposite sides are non-parallel.

Area, A = $\frac{1}{2}$ $\times$ sum of parallel sides $\times$ distance between them

Calculation:

The given solid can be identified as trapezoidal prism. That is it has two congruent trapezoids for its top and lower base. It have four rectangles that connect the corresponding sides of the bases.

1. Area of trapezium at base:

The parallel-sides have length 13 and 20 cm. Also the distance between them is 21 cm.

  $\begin{array}{c}{A}_{\text{base}}=\frac{1}{2}\left(13+20\right)21\\ =\frac{1}{2}\times 33\times 21\end{array}$

  $=346.5$ sq.cm

2. Area of rectangle at bottom:

Length = 20 cm

Breadth = 28 cm

  $\begin{array}{c}{A}_{bottom}=20\times 28\\ =560{\text{cm}}^2\end{array}$

3. Area of rectangle at top:

Length = 13 cm

Breadth = 28 cm

  $\begin{array}{c}{A}_{top}=13\times 28\\ =364{\text{cm}}^2\end{array}$

4. Area of rectangle at left:

Length = 21 cm

Breadth = 28 cm

  $\begin{array}{c}{A}_{left}=21\times 28\\ =588{\text{cm}}^2\end{array}$

5. Area of rectangle at right:

To find the missing edge along the right side draw a vertical line down from the upper right corner as shown below.

  

To find the value of x we can use Pythagoras theorem

That is, $7^2+{21}^2=x^2$

  $x^2=\sqrt{490}$

  $x\approx 22.14$

∴$A_{right}=22.14\times 28$

  $A_{right}=619.8$

Total surface area of the solid is the sum of the areas of the base and area of lateral faces.

  $SA=2\times A_{base}+A_{top}+A_{bottom}+A_{left}+A_{right}$

  $=\left(2\times 346.5\right)+364+560+588+619.8$

  $=2824.8{\text{cm}}^2$

  $$