A company makes biodegradable paper cups in the shape of a right circular cone - Leaving Cert Mathematics - Question 7 - 2020

The curved surface area (CSA) of a cone is calculated using the formula:

$CSA = \pi r l$
Where r is the radius and l is the slant height. Substituting the known values: $CSA = \pi \cdot 3.3 \cdot 9$
Calculating, we find: $CSA \approx 93.31 \, cm^2$
Thus, the curved surface area of the cup is approximately 93.31 cm², correct to 2 decimal places.

To find the angle θ at the apex of the cone, we first calculate the circumference of the base:

$C = 2\pi r = 2\pi \cdot 3.3 \approx 20.74 \, cm$
Next, using the angle subtended by the arc length:

$\text{Arc length} = \frac{θ}{360} \cdot C$
Substituting for arc length (which is the slant height) gives: $9 = \frac{θ}{360} \cdot 20.74$
Solving for θ produces: $θ = \frac{9 \cdot 360}{20.74} \approx 132º$
Therefore, the angle θ is approximately 132 degrees.

The height of water in the cup when filled to line F is: $h = 7.37 \text{ cm}$
To find the radius (r) at this height, we can use similar triangles:

For the full cone: $r : 3.3 = h : 9$
Substituting the values gives: $r = \frac{3.3}{9} \cdot 7.37 \approx 2.905 \, cm$
Now to find the volume (V) of the cone filled with water:

$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (2.905)^2 (7.37) \approx 65.16 \, cm^3$
Thus, the volume of water is approximately 65.2 cm³, correct to 1 decimal place.

To calculate the time (T), we first find the volume rate of flow through the pipe: $\text{Volume flow rate} = \pi r^2 \cdot {flow-rate}$
Substituting the radius of the pipe (0.8 cm) and flow rate (2.5 cm/s): $\text{Volume flow rate} = \pi (0.8)^2(2.5) \approx 5.0265 \, cm^3/s$
Using the volume we found previously (65.16 cm³): $T = \frac{65.16}{5.0265} \approx 13 \, s$
Thus, it will take approximately 13 seconds to fill the cup to the line F.

To determine how far the line F should be drawn to limit the capacity of the cup to 60 cm³, we use the volume formula:

$60 = \frac{1}{3} \pi r^2 h$
We already found r at the new height: $r = \frac{3.3}{9} \cdot h$
Substituting into the volume formula yields: $60 = \frac{1}{3} \pi \left(\frac{3.3}{9}h\right)^2 h$
After simplifying and solving: $h \approx 7.169 \, cm$
Therefore, the line F should be drawn approximately 1.2 cm vertically below the rim of the cup.