### 6.8 Problem Solving: Using Equations

Problem #

Problem Statement

Problems Similar To

1.

The sum of a number and its square is 72. Find the number?

\begin{equation}x^{2}+x=72\end{equation}

2.

A rectangular counter-top has a hole drilled in it to hold a cylindrical container (a utensil holder). The area of the entire counter top is given by (5x+4)*(x+9), while the area of the hole is given by $(x+9)^{2}$. Write an expression for the area (in factored form) of the counter-top that is left after the hole is drilled.

\begin{equation}(5x+4)(x+9)-(x+9)^{2}\end{equation}

3.

Daphne and Stephanie have competing refreshment stand businesses. Daphne's profit can be modeled by the polynomial $x^{2} + 5x + 3$, where 'x' is the number of items sold. Stephanie's profit can be modeled by the polynomial $8x^{2} + 9x + 2$, where 'x' has the same meaning. How much more is Stephanie's profit than that of Daphne's?

\begin{equation}8x^{2}+9x+2-(x^{2}+5x+3)\end{equation}

4.

Area of a circle is 144*$\pi$ sq inches. If the radius is (x - 4), what is the value of 'x'?

\begin{equation}(x-4)^{2}=144\end{equation}

5.

Area of a square is 169. If a side of the square is (2*x - 3) inches long, what is the value of 'x'?

\begin{equation}(2x-3)^{2}=169\end{equation}

6.

A rectangle has one side 10 ft longer than the other, and its area is 96 $ft^{2}$. Find the length of the shorter side of the rectangle.

\begin{equation}(x+10)x=96\end{equation}

7.

A triangular banner has an area of 35/2 $ft^{2}$. The height of the banner is 2 ft longer than its base. Find the base of the triangle.

\begin{equation}\frac{1}{2} \times x(x+2)=\frac{35}{2}\end{equation}