Chapter 8.3, Problem 41P

Multi Step The height of a painting is twice its width x. You want a 3 inch wide wooden frame for the painting. The area of the frame alone is 216 square inches.

a. Draw a diagram that represents this situation.

b. Write a variable expression for the area of the frame alone.

c. What are the dimensions of the frame?

To determine

To graph: The height of a painting is twice of its width $x$ . You want 3 inch wide wooden frame for the painting. The area of the frame alone is 216 square inches. Draw a diagram that represents this situation.

Explanation of Solution

Given:

The height of a painting is twice of its width $x$ and there is 3 inch wide wooden frame for the painting.

Calculation:

It is given that width of the painting is $x$ and height of painting is twice of its width so height of the painting is $2x$ .

  

To determine

To calculate: The height of a painting is twice of its width $x$ . You want 3 inch wide wooden frame for the painting. The area of the frame alone is 216 square inches. Write a variable expression for the area of the frame alone.

Answer to Problem 41P

The expression for the area of frame alone is $18x-36$

Explanation of Solution

Given:

To find the area of frame alone we should subtract the area of smaller rectangle from the area of larger rectangle from the figure shown in above part.

Calculation:

Let $A$ be the area of frame alone, so

  $A=$ Area of Larger Rectangle $-$ Area of Smaller Rectangle

  $A=2x*x-\left(2x-6\right)\left(x-6\right)$

  $A=2x^2-\left(2x^2-12x-6x+36\right)$

  $A=2x^2-2x^2+18x-36$

  $A=18x-36$

Conclusion:

The expression for the area of frame alone is $18x-36$

To determine

To calculate: The height of a painting is twice of its width $x$ . You want 3 inch wide wooden frame for the painting. The area of the frame alone is 216 square inches. What are the dimensions of the frame?

Answer to Problem 41P

The length of frame is 28 inches and the width of frame is 14 inches

Explanation of Solution

Given:

The area of the frame alone is 216 square inches

Calculation:

From the above part we know the expression of the area of frame alone is $18x-36$ and its area is given as 216 square inches.

Now we know the expression for the area of frame alone and its area also so by equating them we get

  $A=18x-36$

  $216=18x-36$

  $216+36=18x$

  $252=18x$

  $x=\frac{252}{18}=14$

Width of frame is 14 inches and its length is twice of width, so

length $=2*14=28$ inches

Conclusion:

The length of frame is 28 inches and the width of frame is 14 inches.