Question

there are 48 blocks for the stack, and each block is 1.5\×2.5\×7.5\.
round all the answers to the nearest tenth, if needed.
a. if elena and her mom stack all 48 blocks to begin the game, as shown above, what is the volume of the stack?
v = type your answer... in³
b. what is the surface area of this original stack?
sa = type your answer... cm²
c. if elena and her mom want to paint each side of each block, how much paint do they need?
sa = type your answer... in²

Explanation:

Step1: Calculate the volume of one block

The volume formula for a rectangular prism is \( V = l\times w\times h \). For one block, \( l = 7.5\) in, \( w = 2.5\) in, \( h = 1.5\) in. So \( V_{1}=7.5\times2.5\times1.5=\frac{15}{2}\times\frac{5}{2}\times\frac{3}{2}=\frac{225}{8}=28.125\) in³.

Step2: Calculate the volume of the stack

There are \( n = 48\) blocks. The volume of the stack \( V=n\times V_{1}\). So \( V = 48\times28.125=1350\) in³.

Step3: Calculate the surface - area of one block

The surface - area formula for a rectangular prism is \( SA=2(lw + lh+wh)\). Substitute \( l = 7.5\) in, \( w = 2.5\) in, \( h = 1.5\) in.

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Step4: Calculate the surface - area for part c

Since they want to paint each side of each block and there are \( n = 48\) blocks. The total surface - area \( SA=n\times SA_{1}\). So \( SA=48\times67.5 = 3240\) in².

For part b, assume the stack is a rectangular prism. If we stack the blocks, assume the number of blocks along the length \( x\), width \( y\), and height \( z\) such that \( xyz=48\). Let's assume a simple stacking: if we stack them in a \( 4\times4\times3\) (length \( l = 4\times7.5 = 30\) in, width \( w=4\times2.5 = 10\) in, height \( h = 3\times1.5=4.5\) in)

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Answer:

a. \( 1350\) in³
b. \( 960\) in²
c. \( 3240\) in²