Question

text to speech protractor calculator incorrect 2 tries left. please try again. the science team is creating a float for the homecoming parade. they want to make a rocket ship by using foam to create a cone that is stacked on top of a cylinder. the bases of the cylinder and cone will be congruent and have a radius of 1.5 feet. the height of the cylinder will be 4 feet, and the height of the cone will be 2 feet. if all the visible surfaces of the structure need to be painted except for the bottom of the cylinder, then what area needs to be painted? round your answer to the nearest tenth of a square foot. 56.5 ft²

Explanation:

Step1: Calculate the lateral surface area of the cylinder

The formula for the lateral surface area of a cylinder is \(2\pi rh\). Here, \(r = 1.5\) feet and \(h = 4\) feet.

$$2\pi\times1.5\times4=12\pi$$

Step2: Calculate the area of one base of the cylinder

The formula for the area of a circle (base of the cylinder) is \(\pi r^{2}\). Here, \(r = 1.5\) feet.

$$\pi\times(1.5)^{2}=2.25\pi$$

Step3: Calculate the slant height of the cone

Use the Pythagorean theorem \(l=\sqrt{r^{2}+h^{2}}\) for the cone. Here, \(r = 1.5\) feet and \(h = 2\) feet.

$$l=\sqrt{(1.5)^{2}+2^{2}}=\sqrt{2.25 + 4}=\sqrt{6.25}=2.5$$

Step4: Calculate the lateral surface area of the cone

The formula for the lateral surface area of a cone is \(\pi rl\). Here, \(r = 1.5\) feet and \(l = 2.5\) feet.

$$\pi\times1.5\times2.5 = 3.75\pi$$

Step5: Calculate the total painted area

Add the lateral surface area of the cylinder, the area of one base of the cylinder, and the lateral surface area of the cone.

$$12\pi+2.25\pi+3.75\pi=(12 + 2.25+3.75)\pi=18\pi$$

Substitute \(\pi\approx3.14\)

$$18\times3.14 = 56.52\approx56.5$$

Answer:

\(56.5\) \(ft^{2}\)