Question

if you add 2 layers to the rectangular prism in problem 4, how much greater is the volume? show your work. what does the prism in problem ask you to find?

Explanation:

Step1: Recall volume formula for rectangular prism

The volume formula for a rectangular prism is $V = l\times w\times h$ (where $l$ is length, $w$ is width and $h$ is height). Adding 2 layers means increasing the height by 2 units (assuming each layer has a height of 1 unit). Let the original height be $h_1$ and the new height be $h_2=h_1 + 2$. The original volume $V_1=l\times w\times h_1$ and the new volume $V_2=l\times w\times h_2=l\times w\times(h_1 + 2)$.

Step2: Find the difference in volumes

The difference $\Delta V=V_2 - V_1=l\times w\times(h_1 + 2)-l\times w\times h_1$. Using the distributive property $a(b + c)=ab+ac$, we have $l\times w\times(h_1 + 2)=l\times w\times h_1+2\times l\times w$. So $\Delta V=(l\times w\times h_1+2\times l\times w)-l\times w\times h_1 = 2\times l\times w$. This means the increase in volume is equal to 2 times the area of the base of the rectangular - prism.

Answer:

The increase in volume is 2 times the area of the base of the rectangular prism.