A rectangular picture measures 6 inches by 8 inches. Build frame that takes no more than 100 sq ft. Write equation or inequality for max width.

Question

(x+6)(x+8)=100.  Double distribution leads to x^2+6x+8x+48=100.  Subtract 100 from both sides gets me to x^2+14x-52=0.  What do I do now?

Answer 1

Your equation is incorrect. The frame will have an equal border width around the picture.

 

Let x be the width of the border around the picture.  Then the area of the frame is

 

(x + 6 + x)(x + 8 + x) ≤ 100

 

(2x + 6)(2x + 8) ≤ 100

 

 

Distribute on the left side.

 

4x² + 24x + 48 ≤ 100

 

 

Subtract 100 on both sides of the equality.

 

4x² + 24x - 52 ≤ 0

 

 

Now solve the inequality for x.  Factor the left side.

 

4(x² + 6x - 13) ≤ 0

 

 

Use the quadratic formula to find the zeros.  Your maximum width will be the greater zero you find.