Questions: Grade 5 Using Models to Explore x +; Lesson 9 Standard NC.S.MD. 5 Special Delivery Ms. Hale ordered supplies for her new classroom. According to the information she uses to track the shipping, the combined volume of her packages is 210 cubic feet. When the packages (in the shape of rectangular prisms) arrive, she finds them stacked on her front porch and challenges her son to find the volume and dimensions of the separate prisms using only the information she gives him. Use the following information to determine the volume and dimensions of each package: - The side lengths are all whole numbers. - The packages have the same height. - The volume of Package 2 is half the volume of Package 1. - Package 1 has a base of 20 square units.

Solution Steps

To solve this problem, we need to determine the volume and dimensions of each package based on the given information. Here's the approach:

1. Identify the volume of Package 1: Since the total volume is 210 cubic feet and Package 2 is half the volume of Package 1, we can set up an equation to find the volume of Package 1.
2. Calculate the volume of Package 2: Once we have the volume of Package 1, we can easily find the volume of Package 2.
3. Determine the height of the packages: Given that Package 1 has a base of 20 square units, we can use the volume to find the height.
4. Find the dimensions of Package 2: Using the height and the volume of Package 2, we can determine its base area and dimensions.

Given the total volume of both packages is \( V_{\text{total}} = 210 \) cubic feet, and the volume of Package 2 is half the volume of Package 1, we can express the volumes as follows:

\[ V_1 + V_2 = V_{\text{total}} \quad \text{and} \quad V_2 = \frac{1}{2} V_1 \]

Substituting \( V_2 \) into the total volume equation:

\[ V_1 + \frac{1}{2} V_1 = 210 \]

This simplifies to:

\[ \frac{3}{2} V_1 = 210 \]

Solving for \( V_1 \):

\[ V_1 = 210 \times \frac{2}{3} = 140 \text{ cubic feet} \]

Using the volume of Package 1, we can find the volume of Package 2:

\[ V_2 = \frac{1}{2} V_1 = \frac{1}{2} \times 140 = 70 \text{ cubic feet} \]

Given that Package 1 has a base area of \( 20 \) square units, we can find the height \( h_1 \) using the formula for volume:

\[ V_1 = \text{Base Area} \times \text{Height} \implies 140 = 20 \times h_1 \]

Solving for \( h_1 \):

\[ h_1 = \frac{140}{20} = 7 \text{ feet} \]

Assuming the dimensions of Package 1 are \( \text{length} = 20 \) feet and \( \text{width} = 1 \) foot, we have:

\[ \text{Dimensions of Package 1} = 20 \times 1 \times 7 \]

For Package 2, we know its volume and height. The base area \( A_2 \) can be calculated as follows:

\[ A_2 = \frac{V_2}{h_1} = \frac{70}{7} = 10 \text{ square units} \]

Assuming the dimensions of Package 2 are \( \text{length} = 10 \) feet and \( \text{width} = 1 \) foot, we have:

\[ \text{Dimensions of Package 2} = 10 \times 1 \times 7 \]

Final Answer