Question

select the correct answer
a waffle is rolled to make a cone - shape without any gaps or overlaps. the radius of its base is 2.5 centimeters, and its slant height is 11.5 centimeters. the company wants to save money on the cone by reducing the size.
which change would result in the biggest reduction of the lateral surface area of the cone?
a. decrease radius by 0.3 cm and slant height by 0.2 cm
b. decrease slant height by 0.5 cm
c. decrease radius by 0.5 cm
d. decrease radius by 0.2 cm and slant height by 0.3 cm

Explanation:

Step1: Recall lateral - surface area formula

The lateral - surface area of a cone is given by $A = \pi rl$, where $r$ is the radius and $l$ is the slant height. The original radius $r_0=2.5$ cm and the original slant height $l_0 = 11.5$ cm, so the original lateral - surface area $A_0=\pi\times2.5\times11.5 = 28.75\pi$ $cm^2$.

Step2: Calculate new areas for option A

The new radius $r_A=2.5 - 0.3=2.2$ cm and the new slant height $l_A=11.5 - 0.2 = 11.3$ cm. The new lateral - surface area $A_A=\pi\times2.2\times11.3 = 24.86\pi$ $cm^2$. The reduction in area $\Delta A_A=28.75\pi-24.86\pi = 3.89\pi$ $cm^2$.

Step3: Calculate new areas for option B

The new slant height $l_B=11.5 - 0.5 = 11$ cm. The new lateral - surface area $A_B=\pi\times2.5\times11 = 27.5\pi$ $cm^2$. The reduction in area $\Delta A_B=28.75\pi - 27.5\pi=1.25\pi$ $cm^2$.

Step4: Calculate new areas for option C

The new radius $r_C=2.5 - 0.5 = 2$ cm. The new lateral - surface area $A_C=\pi\times2\times11.5 = 23\pi$ $cm^2$. The reduction in area $\Delta A_C=28.75\pi-23\pi = 5.75\pi$ $cm^2$.

Step5: Calculate new areas for option D

The new radius $r_D=2.5 - 0.2 = 2.3$ cm and the new slant height $l_D=11.5 - 0.3 = 11.2$ cm. The new lateral - surface area $A_D=\pi\times2.3\times11.2 = 25.76\pi$ $cm^2$. The reduction in area $\Delta A_D=28.75\pi-25.76\pi = 2.99\pi$ $cm^2$.

Answer:

C. decrease radius by 0.5 cm