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 the height of the cylinder

The volume formula of a cylinder is \(V=\pi r^{2}h\). Given \(V = 350\space cm^{3}\), \(r = 3\space cm\), we can solve for \(h\).

$$h=\frac{V}{\pi r^{2}}=\frac{350}{\pi\times3^{2}}=\frac{350}{9\pi}\approx\frac{350}{9\times3.14}\approx 12.4\space cm$$

Step2: Calculate the height of the rectangular prism

The volume formula of a rectangular prism is \(V = l\times w\times h\). Given \(V=350\space cm^{3}\), \(l = 6\space cm\), \(w = 4\space cm\), we solve for \(h\).

$$h=\frac{V}{l\times w}=\frac{350}{6\times4}=\frac{350}{24}\approx14.6\space cm$$

Step3: Calculate the surface area of the cylinder

The surface area formula of a cylinder is \(S_{cylinder}=2\pi r(r + h)\). Substitute \(r = 3\space cm\), \(h\approx12.4\space cm\)

$$S_{cylinder}=2\pi\times3\times(3 + 12.4)=6\pi\times15.4\approx6\times3.14\times15.4 = 290.5\space cm^{2}$$

Step4: Calculate the surface area of the rectangular prism

The surface area formula of a rectangular prism is \(S_{prism}=2(lw+lh + wh)\). Substitute \(l = 6\space cm\), \(w = 4\space cm\), \(h\approx14.6\space cm\)

$$S_{prism}=2(6\times4+6\times14.6+4\times14.6)=2(24 + 87.6+58.4)=2\times170=340\space cm^{2}$$

Answer:

a. The height of the cylinder is approximately \(12.4\space cm\)
b. The height of the rectangular prism is approximately \(14.6\space cm\)
c. The cylinder requires less surface area.