### 6.8 Problem Solving: Using Equations

Problem #

Problem Statement

Problems Similar To

1.

The sum of a number and its square is 42. Find the number?

$$x^{2}+x=42$$

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 (10x+6)*(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.

$$(10x+6)(x+9)-(x+9)^{2}$$

3.

Daphne and Stephanie have competing refreshment stand businesses. Daphne's profit can be modeled by the polynomial $x^{2} + 4x + 3$, where 'x' is the number of items sold. Stephanie's profit can be modeled by the polynomial $2x^{2} + 6x + 5$, where 'x' has the same meaning. How much more is Stephanie's profit than that of Daphne's?

$$2x^{2}+6x+5-(x^{2}+4x+3)$$

4.

Area of a circle is 324*$\pi$ sq inches. If the radius is (x - 9), what is the value of 'x'?

$$(x-9)^{2}=324$$

5.

Area of a square is 121. If a side of the square is (7*x - 9) inches long, what is the value of 'x'?

$$(7x-9)^{2}=121$$

6.

A rectangle has one side 8 ft longer than the other, and its area is 153 $ft^{2}$. Find the length of the shorter side of the rectangle.

$$(x+8)x=153$$

7.

A triangular banner has an area of 75/2 $ft^{2}$. The height of the banner is 10 ft longer than its base. Find the base of the triangle.

$$\frac{1}{2} \times x(x+10)=\frac{75}{2}$$