Nicky wants to surround an 11 in. by 5 in. painting with a frame that is w inches wide. Which quadratic function represents the combined area A of the picture and the frame?

Question

I am struggling with problem, I can't find the answer. So if someone could explain it to me it would be nice. Thank you.

Answer 1

Since the frame is uniform around the picture, you need to add its width, W, to the sides of the picture. We will manipulate the formula for area, A = L x W. To ease confusion with the chosen variable, I will replace the W with an H for height to be A = L x H.

L = W + 11 + W

H = W + 5 + W

You can see this by drawing a picture with a frame around it. Looking from the top to the bottom, you will have it's width plus the picture plus the width of the frame again. Same thing for the height of the picture. Looking from the far left of the frame and across, you have the width of the frame plus the picture plus the width of the frame again. Now plug these into the area formula and combine like terms.

A = L x H

A = (W + 11 + W) x (W + 5 +W)

A = (2W + 11) x (2W + 5)

Now take the product of the two parenthesis.

I learned to use the FOIL method: multiply First, Outside, Inside, Last.

A = (2W x 2W) + (2W x 5) + (11 x 2W) + (11 x 5)

A = 4W² + 10W + 22W + 55

A = 4W² + 32W + 55

Answer 2

the length of the painting is 11 inches

the width of the painting is 5 inches

the frame is w inches long all the way around

the width of the painting becomes w=5+w+w=2w+5 (w inches wider on both sides)

the length of the painting becomes l=11+w+w=2w+11( w inches longer on the top and on the bottom)

A=l*w

A=(2w+11)(2w+5)

A=4w^2+22w+10w+55

A=4w^2+32w+55

Answer 3

The dimensions of the rectangular area after the painting is surrounded by a frame of width "w" will be 2w+11 and 2w+5. This is because there will be a "w" added to all 4 sides of the painting.

The area of the rectangle is given as A = lw or in this case: A = (2w+11)(2w+5). If this question is multiple choice you might have to multiply those two binomials together (FOIL) to arrive at the answer provided.