Question

1. exploration

an object has volume 36 cubic inches. a box has side lengths 1 foot by 3 inches by 4 inches.
a. what is the smallest number of these objects that can fit in the box? explain your reasoning.
b. what is the largest number of these objects that can fit in the box? explain your reasoning.

1. exploration

a container has a volume of 120 cubic inches.
a. what could the length, width, and height of the container be?
b. can one of the side lengths be 9 inches? explain or show your reasoning.

Explanation:

6a.

Step1: Convert box dimensions to inches

1 foot = 12 inches, so box dimensions are 12 inches by 3 inches by 4 inches. Volume of box $V = 12\times3\times4=144$ cubic - inches.

Step2: Calculate smallest number

If the objects have irregular shapes and do not pack efficiently, we consider the worst - case scenario. If the objects do not fit neatly, we might not be able to fill the box completely. But if we assume we can start filling, the smallest non - zero number of objects that can fit is 1.

6b.

Step1: Calculate volume of box

As calculated before, volume of box $V = 12\times3\times4 = 144$ cubic inches.

Step2: Calculate largest number

To find the largest number of objects that can fit, we assume they pack perfectly. Let $n$ be the number of objects. $n=\frac{V_{box}}{V_{object}}=\frac{144}{36}=4$.

7a.

Step1: Recall volume formula

The volume of a rectangular prism is $V = l\times w\times h$, where $l$ is length, $w$ is width, and $h$ is height. We need to find three positive numbers such that $l\times w\times h=120$. Some possible combinations are: $l = 10$ inches, $w = 6$ inches, $h = 2$ inches (since $10\times6\times2 = 120$).

7b.

Step1: Assume a side length

Let's assume one side length $l = 9$ inches. Then $w\times h=\frac{120}{9}=\frac{40}{3}\approx13.33$. Since $w$ and $h$ are positive real numbers, it is possible to find values for $w$ and $h$ such that $w\times h=\frac{40}{3}$. For example, $w = 4$ inches and $h=\frac{10}{3}$ inches. So one of the side lengths can be 9 inches.

Answer:

6a. The smallest number of objects is 1.
6b. The largest number of objects is 4.
7a. One possible set of side - lengths: length = 10 inches, width = 6 inches, height = 2 inches.
7b. Yes, one of the side lengths can be 9 inches.