A painting without its frame has a width 2.5 times its height, h, in inches. Its frame is 3 in. wide all along its perimeter. What is an expression for the area of the framed painting in terms of h?

Question

A. 2.5h²

B. 2.5h² + 6

C. 2.5h² + 36

D. 2.5h² + 21h + 36

Answer 1

Hi Anastasia,

Ok let's break down the given information. We have a painting without frame which is a rectangle with width and height. Recall the equations for area and perimeter of a rectangle.

A = h*w

P = 2h + 2w

Now, it says the frame adds an extra 3 inches wide on all sides along the perimeter. This means the frame adds 3 inches on the left and 3 inches on the right, so the overall width increases by 3+3=6 inches. The frame addes 3 inches on top and 3 inches on bottom, so the overall height increases by 3+3=6 inches.

Given:

w = 2.5*h

Since the h and w represent the height and width of painting without frame, we need to add in the 6 inches of the frame in the area equation:

A = (h+6)*(w+6)

Now the question is asking for area in terms of h, so let's substitute (2.5*h) in place of w:

A = (h+6)*(2.5*h+6)

Simplify:

A = 2.5h²+6h+15h+36

A = 2.5h²+21h+36

Answer is D.

Hope this helps! :)

Answer 2

Ok so here we’re dealing with finding the area of a framed painting. We’re given the dimensions of the painting and then given the added dimensions of the frame. Area will be calculated using width w multiplied by height h. So we know w=2.5h for just the painting but we want the width of the framed painting so that total width will be

W=2.5h+6

since we have to add two lengths of 3in to the width, and the height of the framed painting will be the height of the painting plus two lengths of 3in on top and bottom so the total height will be

H=h+6

Then the desired area will be the product of width times height.

A=HW=(2.5h+6)(h+6)=2.5h²+21h+36

Always helps to draw a picture to see what you’re working with