Question

1. application problem

a company wants to design a container that holds 350 cm³ of liquid.
they consider two shapes:
· cylinder: radius 3 cm
· rectangular prism: base 4 cm × 6 cm
a. what height should the cylinder have to reach 350 cm³?
b. what height should the rectangular prism have for the same volume?
c. which shape requires less surface area?
show calculations.

Explanation:

Step1: Calculate height of cylinder

The volume formula for a cylinder is \(V=\pi r^{2}h\). Given \(V = 350\space cm^{3}\), \(r = 3\space cm\).
We can solve for \(h\) by \(h=\frac{V}{\pi r^{2}}\).
Substitute the values: \(h=\frac{350}{\pi\times3^{2}}=\frac{350}{9\pi}\approx\frac{350}{9\times3.14}\approx12.3\space cm\).

Step2: Calculate height of rectangular prism

The volume formula for a rectangular prism is \(V = l\times w\times h\). Given \(V=350\space cm^{3}\), \(l = 4\space cm\), \(w = 6\space cm\).
We solve for \(h\) by \(h=\frac{V}{l\times w}\).
Substitute the values: \(h=\frac{350}{4\times6}=\frac{350}{24}\approx14.6\space cm\).

Step3: Calculate surface area of cylinder

The surface area formula for a cylinder is \(S = 2\pi r(r + h)\). Substitute \(r = 3\space cm\), \(h\approx12.3\space cm\).
\(S=2\pi\times3\times(3 + 12.3)=6\pi\times15.3\approx6\times3.14\times15.3 = 288.5\space cm^{2}\).

Step4: Calculate surface area of rectangular prism

The surface area formula for a rectangular prism is \(S=2(lw+lh + wh)\). Substitute \(l = 4\space cm\), \(w = 6\space cm\), \(h\approx14.6\space cm\).
\(S=2(4\times6+4\times14.6+6\times14.6)=2(24 + 58.4+87.6)=2\times170=340\space cm^{2}\).

Answer:

a. The height of the cylinder is approximately \(12.3\space cm\).
b. The height of the rectangular prism is approximately \(14.6\space cm\).
c. The cylinder requires less surface area.