Question

if the height of the rectangular prisms changes from 5in to 10 in what would the difference in surface area be between the two rectangular prisms? 80 in.² 105 in.² 110 in.² 90 in.²

Explanation:

Step1: Recall surface area formula for rectangular prism

The surface area \( S \) of a rectangular prism with length \( l \), width \( w \), and height \( h \) is given by \( S = 2(lw + lh + wh) \). From the diagram, we can assume the length \( l = 4 \) in, width \( w = 7 \) in (since the base has dimensions 4 in and 7 in, and the height is changing). Let's denote the initial height as \( h_1 = 5 \) in and the new height as \( h_2 = 10 \) in.

Step2: Calculate surface area for \( h_1 = 5 \) in

Substitute \( l = 4 \), \( w = 7 \), \( h_1 = 5 \) into the formula:
\[

$$\begin{align*} S_1&=2(4\times7 + 4\times5 + 7\times5)\\ &=2(28 + 20 + 35)\\ &=2(83)\\ &= 166 \text{ in}^2 \end{align*}$$

\]

Step3: Calculate surface area for \( h_2 = 10 \) in

Substitute \( l = 4 \), \( w = 7 \), \( h_2 = 10 \) into the formula:
\[

$$\begin{align*} S_2&=2(4\times7 + 4\times10 + 7\times10)\\ &=2(28 + 40 + 70)\\ &=2(138)\\ &= 276 \text{ in}^2 \end{align*}$$

\]

Step4: Find the difference in surface areas

The difference \( \Delta S=S_2 - S_1 \). So, \( \Delta S=276 - 166 = 110 \text{ in}^2 \)? Wait, no, wait. Wait, maybe I misread the width. Wait, maybe the base is length 4, width 4? No, the diagram shows 4 in, 7 in, and 5 in. Wait, maybe the width is 4? Wait, no, let's re - examine. Wait, maybe the length is 4, width is 4? No, the problem says rectangular prisms. Wait, maybe the initial dimensions: from the diagram, the first prism (upper) has height 5 in, and the lower one has height 7 in, and base 4 in. Wait, maybe the length \( l = 4 \), width \( w = 4 \)? No, that can't be. Wait, maybe the correct dimensions: let's assume that the length \( l = 4 \), width \( w = 7 \) (the base of the lower prism is 4 in and 7 in), and the height is the vertical side. Wait, maybe I made a mistake in the width. Wait, let's re - derive the formula for the difference in surface area.

The surface area formula is \( S = 2(lw+lh + wh)=2lw + 2lh+2wh \). When the height changes from \( h_1 \) to \( h_2 \), the change in surface area comes from the terms involving \( h \). The terms with \( lw \) are constant (since \( l \) and \( w \) don't change). So the difference in surface area \( \Delta S=2l(h_2 - h_1)+2w(h_2 - h_1)=2(l + w)(h_2 - h_1) \).

Given \( l = 4 \), \( w = 7 \), \( h_2 - h_1=10 - 5 = 5 \).

Then \( \Delta S=2(4 + 7)\times5=2\times11\times5 = 110 \text{ in}^2 \)? Wait, but let's check again. Wait, if \( l = 4 \), \( w = 4 \)? No, the diagram shows 4 in, 7 in, 5 in. Wait, maybe the length is 4, width is 7, height changes. Let's recalculate the difference using the formula for the change.

The surface area of a rectangular prism is \( S = 2(lw+lh + wh) \). The difference between the two surface areas when height changes from \( h_1 \) to \( h_2 \) is:

\( \Delta S=2(lw+lh_2 + wh_2)-2(lw+lh_1 + wh_1)=2l(h_2 - h_1)+2w(h_2 - h_1)=2(l + w)(h_2 - h_1) \)

We know that \( h_2 - h_1=10 - 5 = 5 \), \( l = 4 \), \( w = 7 \) (from the diagram, the base has length 4 and width 7, as the lower prism has height 7 and base 4). Then:

\( \Delta S=2(4 + 7)\times5=2\times11\times5 = 110 \text{ in}^2 \)

Answer:

\( 110 \text{ in}^2 \) (corresponding to the option "110 in.²")