Question

1. a company is designing a new cylindrical water bottle. the volume of the bottle is 103 cubic centimeters. what is the radius of the water bottle? estimate using 3.14 for π, and round to the nearest hundredth.

(there is an image of a water bottle with a height labeled 8.1 cm at the bottom)

Explanation:

Step1: Recall Cylinder Volume Formula

The volume \( V \) of a cylinder is given by \( V=\pi r^{2}h \), where \( r \) is the radius and \( h \) is the height. We know \( V = 103\space cm^{3}\), \( \pi=3.14 \), and from the diagram, \( h = 8.1\space cm \). We need to solve for \( r \).

Step2: Rearrange the Formula for \( r \)

Starting with \( V=\pi r^{2}h \), divide both sides by \( \pi h \) to get \( r^{2}=\frac{V}{\pi h} \). Then take the square root: \( r=\sqrt{\frac{V}{\pi h}} \).

Step3: Substitute the Values

Substitute \( V = 103 \), \( \pi=3.14 \), and \( h = 8.1 \) into the formula:
\( r^{2}=\frac{103}{3.14\times8.1} \). First calculate the denominator: \( 3.14\times8.1 = 25.434 \). Then \( r^{2}=\frac{103}{25.434}\approx4.05 \).

Step4: Find the Radius

Take the square root of \( 4.05 \): \( r=\sqrt{4.05}\approx2.01 \) (rounded to the nearest hundredth).

Answer:

The radius of the water bottle is approximately \( 2.01 \) centimeters.