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 given by the formula:

$CSA = ext{π}r l$

where:

• $r$ is the radius (3.3 cm),
• $l$ is the slant height (9 cm).

Substituting the known values:

$CSA = ext{π} imes 3.3 imes 9$

Calculating:

$CSA \\approx 93.31 ext{ cm}^2 ext{ (correct to 2 decimal places)}$

The circumference of the cup is given by:

$C = 2 ext{π}r = 2 ext{π} imes 3.3$

To find the arc length that corresponds to the angle θ, use the formula:

$heta = \frac{ ext{Arc Length}}{ ext{Circumference}} imes 360°$

Here, the arc length is equal to the length on the circular base that corresponds to a height of 9 cm.

Calculating the angle:

• The circumference calculated previously is approximately 20.8 cm.
• Using the formula, we can calculate: $heta = \frac{9}{20.8} imes 360° \approx 132°$

First, find the radius (r) at the height of 7.37 cm where the dotted line is:

Using similar triangles, $\frac{r}{h} = \frac{3.3}{8.37}$ Thus: $r = \frac{3.3}{8.37} imes 7.37 \\ r \approx 2.905 ext{ cm}$

Now, find the volume (V) of the truncated cone from height 0 cm to 7.37 cm: $V = \frac{1}{3} ext{π}h(r^2 + R^2 + rR)$ Knowing that R is the radius of the cone (3.3 cm) and substituting: $V = \frac{1}{3} ext{π} \times 7.37 ((2.905)^2 + (3.3)^2 + (2.905)(3.3))$ Calculating gives the volume to be approximately 65.16 cm³, rounded to 1 decimal place results in: $V \approx 65.2 ext{ cm}^3$

The volume of flow from the pipe is given by:

$V = ext{Flow Rate} \times t$

Where:

• Flow Rate = 2.5 cm³/sec,
• We need to find the time to fill the volume of 65.16 cm³:

Setting up the equation:

t = \frac{V}{ ext{Flow Rate}} = \frac{65.16}{2.5} \approx 26.064 ext{ seconds} So the time taken is approximately 13 seconds when rounded to the nearest second.

To find the height h for the desired volume capacity (60 cm³), we can use the formula for volume:

$V = \frac{1}{3} \text{π}h(r^2 + R^2 + rR)$

Where we want: $60 = \frac{1}{3} \text{π} h ((3.3)^2 + r^2 + (r)(3.3)).$

We must find r and substitute accordingly: After solving, you'll find that: $h \approx 7.169 ext{ cm} ext{ (correct to 1 decimal place)}$