Question

shared task set
mayur is in charge of creating themed popcorn containers for a county fair. he plans on creating cylindrical and cone - shaped containers. each cylindrical container, x, has a volume of 52 in³, and each cone - shaped container, y, has a volume of 26 in³. both containers have a height of 5.5 in. the system of equations shown can be used to find the number of cylindrical and cone - shaped containers at each snack booth in the fair.
52x + 26y = 7150
x + y = 175
given that the diameter of the cone - shaped container is 4.25 in, find the slant height of the cone - shaped containers. round your answer to the nearest tenth.
a 7.0 in
b 5.1 in
c 5.9 in
d 7.6 in

Explanation:

Step1: Recall the volume formula for a cone

The volume of a cone is given by \( V = \frac{1}{3}\pi r^2 h \), where \( r \) is the radius and \( h \) is the height. We know the volume \( V = 26 \, \text{in}^3 \) and the height \( h = 5.5 \, \text{in} \). First, we need to find the radius \( r \).

Step2: Solve for the radius \( r \)

From the volume formula:
\[
26=\frac{1}{3}\pi r^{2}(5.5)
\]
Multiply both sides by 3:
\[
78 = 5.5\pi r^{2}
\]
Divide both sides by \( 5.5\pi \):
\[
r^{2}=\frac{78}{5.5\pi}\approx\frac{78}{17.28}\approx4.51
\]
Take the square root:
\[
r\approx\sqrt{4.51}\approx2.12 \, \text{in}
\]
The diameter \( d = 2r \), so \( d\approx4.24 \, \text{in} \) (wait, but the problem says the diameter is 4.25 in, maybe a typo or approximation. Now, to find the slant height \( l \) of the cone, we use the Pythagorean theorem for the cone: \( l=\sqrt{r^{2}+h^{2}} \), where \( r \) is the radius (diameter is 4.25, so radius \( r=\frac{4.25}{2} = 2.125 \, \text{in} \)) and \( h = 5.5 \, \text{in} \).

Step3: Calculate the slant height

\[
l=\sqrt{(2.125)^{2}+(5.5)^{2}}=\sqrt{4.515625 + 30.25}=\sqrt{34.765625}\approx5.9 \, \text{in}
\]

Answer:

C. 5.9 in