Question

d. if the box originally had 5000 staples and jacques estimated that he needs 2000 staples to finish the job, does he have enough?

1. find the width of the rectangular prism with a height of 1.5 ft, a length of 3 ft and a volume of 11.25 ft³.
2. a student calculated the radius of the cylinder with a height of 9 inches and a volume of 200 cubic inches.

below is his work:
step 1: v = πr²h
step 2: 200 in³ = πr²(9 in)
step 3: $\frac{200\text{ in}^3}{pi(9\text{ in})}=r^{2}$
step 4: 7.07355 in² = r²
step 5: 3.5 in = r
a. in which step was an error made?
b. what was the error that the student made?

Explanation:

Step1: Recall volume formula for rectangular prism

The volume formula for a rectangular prism is $V = l\times w\times h$, where $V$ is volume, $l$ is length, $w$ is width and $h$ is height. We need to solve for $w$.

Step2: Rearrange the formula for $w$

We can rewrite the formula as $w=\frac{V}{l\times h}$.

Step3: Substitute given values

Given $V = 11.25$ ft³, $l = 3$ ft and $h=1.5$ ft. Then $w=\frac{11.25}{3\times1.5}$.

Step4: Calculate the value of $w$

First, calculate $3\times1.5 = 4.5$. Then $w=\frac{11.25}{4.5}=2.5$ ft.

For the cylinder - error analysis:

Step1: Analyze step 5

The error is in step 5. When we have $r^{2}=7.07355$ in², to find $r$, we should take the square - root of both sides. $r=\sqrt{7.07355}\approx 2.66$ in, not $3.5$ in.

Answer:

1. The width of the rectangular prism is $2.5$ ft.
2. a. Step 5

b. The student incorrectly took the square - root of $7.07355$ in². The correct value of $r$ is $r = \sqrt{7.07355}\approx2.66$ in, not $3.5$ in.