Chapter 9.8, Problem 35PPE

(a)

To determine

To find the area of rectangle.

Answer to Problem 35PPE

  $7.4$ Square units.

Explanation of Solution

Given:

The rectangles are centered on the y-axis with bases on the x-axis.

The upper vertices of the rectangles are defined by the following function;

  $y=-0.3x^2+4$

Calculation:

Draw the following graph.

First find the length and breadth of the rectangle.

The base of the rectangle is $2$ units on the graph; hence the breadth of the rectangle is $2$ units.

Suppose the length of the rectangle is $l$ .

Therefore the upper vertices will be $(-1,l)$ and $(1,l)$

The upper vertices are defined by the function;

  $\begin{array}{}$ y=-0.3x^2+4$$ y=-0.3{(1)}^2+4$$ y=-0.3+4$$ y=3.7$\end{array}$

Therefore the length of the rectangle is $3.7$ units.

Area of a rectangle with length l and breadth b is given as follows:

  $\begin{array}{l}l\times b\\ =3.7\times 2\\ =7.4\end{array}$

Therefore area of the rectangle is $7.4$ square units.

(b)

To determine

To find the area of rectangle.

Answer to Problem 35PPE

  $7.8$ Square units.

Explanation of Solution

Given:

The rectangles are centered on the y-axis with bases on the x-axis.

The upper vertices of the rectangles are defined by the following function;

  $y=-0.3x^2+4$

Calculation:

Draw the following graph.

First find the length and breadth of the rectangle.

The base of the rectangle is $6$ units on the graph; hence the length of the rectangle is $6$ units.

Suppose the breadth of the rectangle is $b$ .

Therefore the upper vertices will be $(-3,b)$ and $(3,b)$

The upper vertices are defined by the function;

  $\begin{array}{}$ y=-0.3x^2+4$$ y=-0.3{(3)}^2+4$$ y=-2.7+4$$ y=1.3$\end{array}$

Therefore the breadth of the rectangle is $1.3$ units.

Area of a rectangle with length l and breadth b is given as follows:

  $\begin{array}{l}l\times b\\ =6\times 1.3\\ =7.8\end{array}$

Therefore area of the rectangle is $7.8$ square units.

(c)

To determine

To find the vertices of the square constructed in the same manner.

Answer to Problem 35PPE

  $(-1.61,0),(1.61,0),(-1.61,3.22),(1.61,3.22)$

Explanation of Solution

Given:

The rectangles are centered on the y-axis with bases on the x-axis.

The upper vertices of the rectangles are defined by the following function;

  $y=-0.3x^2+4$

Calculation:

The square will be constructed as follows-

The following will be the quadratic equation for the $0.3s^2+2s-4=0$ and solving for $s$ , the following vertices of square are obtained-

  $(-1.61,0),(1.61,0),(-1.61,3.22),(1.61,3.22)$

(d)

To determine

To find the area of square rounding to nearest hundredth.

Answer to Problem 35PPE

  $10.37$ square units.

Explanation of Solution

Given:

The rectangles are centered on the y-axis with bases on the x-axis.

The upper vertices of the rectangles are defined by the following function;

  $y=-0.3x^2+4$

Calculation:

The area of square can be found using the graph from part (c).

From part (c), the length of the side of the square is $3.22$ units.

Area of a square with side length a units is given as follows:

  $\begin{array}{}$ A={(3.22)}^2$$ A\approx 10.37$\end{array}$

Therefore the area of the square is approximately $10.37$ square units.