Chapter SH, Problem 4.8EP

a.

To determine

Find the quadratic equation which represents the Julio platform area.

Answer to Problem 4.8EP

The quadratic equation form of the area (new platform) is $x^2+8x-20=0$ .

Explanation of Solution

Given:

The original platform dimension is $3$ feet by $5$ feet. Julio adds $x$ feet to each side and made his platform. The area of the new platform is $35$ square feet.

Calculation:

Length of the original platform $=$$5$ feet

Length of the Julio’s platform $=$$\left(5+x\right)$ feet

Width of the original platform $=$$3$ feet

Width of the Julio’s platform $=$$\left(3+x\right)$ feet

The given diagram is

  

Now, the area of the Julio’s platform $\left(A_J\right)=\left(5+x\right)\left(3+x\right)$ square feet.

The quadratic equation form of the area is

  $\begin{array}{l}35=\left(5+x\right)\left(3+x\right)\\ \Rightarrow 35=15+5x+3x+x^2\\ \Rightarrow 35=15+8x+x^2\end{array}$

Subtract $35$ from both sides.

  $\begin{array}{l}\Rightarrow 35-35=15+8x+x^2-35\\ \Rightarrow 0=8x+x^2-20\\ \Rightarrow x^2+8x-20=0\end{array}$

Hence the quadratic equation form of the area (new platform) is $x^2+8x-20=0$

b.

To determine

Find the dimension of the Julio’s platform.

Answer to Problem 4.8EP

The Julio’s platform dimensions of $5$ feet by $7$ feet.

Explanation of Solution

Given:

The original platform dimensionsare $3$ feet by $5$ feet. Julio adds $x$ feet to each side and made his platform. The area of the new platform is $35$ square feet.

Calculation:

Length of the original platform $=$$5$ feet

Length of the Julio’s platform $=$$\left(5+x\right)$ feet

Width of the original platform $=$$3$ feet

Width of the Julio’s platform $=$$\left(3+x\right)$ feet

The given diagram is

  

Now, the area of the Julio’s platform $\left(A_J\right)=\left(5+x\right)\left(3+x\right)$ square feet.

  $\begin{array}{l}35=\left(5+x\right)\left(3+x\right)\\ \Rightarrow 35=15+5x+3x+x^2\\ \Rightarrow 35=15+8x+x^2\end{array}$

Subtract $35$ from both sides.

  $\begin{array}{l}\Rightarrow 35-35=15+8x+x^2-35\\ \Rightarrow 0=8x+x^2-20\\ \Rightarrow x^2+8x-20=0\\ \Rightarrow x^2+10x-2x-20=0\\ \Rightarrow x\left(x+10\right)-2\left(x+10\right)=0\\ \Rightarrow \left(x+10\right)\left(x-2\right)=0\\ \Rightarrow x+10=0\\ \Rightarrow x+10-10=0-10\\ \Rightarrow x=-10\left[\text{negative value is not possible}\right]\\ \text{and}\\ \Rightarrow x-2=0\\ \Rightarrow x-2+2=0+2\\ \Rightarrow x=2\text{feet}\end{array}$

Length of the Julio’s platform $=$$\left(5+x\right)=5+2=7$ feet

Width of the Julio’s platform $=$$\left(3+x\right)=3+2=5$ feet

Hence the Julio’s platform dimensions of $5$ feet by $7$ feet.