The diagram below is a scale drawing of a hopper tank used to store grain - Leaving Cert Mathematics - Question 9 - 2014

Given that the average adult man is 1.8 m tall and that in the above figure the man is 6 squares tall, it can be inferred that each square represents 0.3 m. Counting the number of squares:

• Height of the hopper tank: Approximately 2.7 m
• Diameter of the tank: Approximately 2.4 m These dimensions can be recorded in the provided spaces on the diagram.

To find the total volume of the hopper tank, we start by calculating the volume of the parts separately:

1. Volume of the cylinder is given by the formula: $V_{cylinder} = ext{Area} imes ext{Height} = \\pi r^2 h \\text{ where } r = 1.2 m, h = 0.9 m$

   Thus, the volume of the cylinder is:
   $V_{cylinder} = \\pi (1.2)^2(0.9) = 4.523 m^3$

2. For the upper cone: $V_{cone_{ ext{top}}} = \frac{1}{3} \\pi r^2 h$

   $V_{cone_{ ext{top}}} = \frac{1}{3}(\pi)(1.2)^2(0.9) = 0.432 m^3$

3. For the lower cone: Using a similar method: $V_{cone_{ ext{bottom}}} = \frac{1}{3} \\pi r^2 h$

   =
   $\frac{1}{3}(\pi)(1.2)^2(0.9) = 0.792 m^3$

4. Finally, the total volume of the hopper tank is: $V_{total} = V_{cylinder}+ V_{cone_{ ext{top}}} + V_{cone_{ ext{bottom}}} = 5.112 m^3$

   Rounded to two decimal places, the capacity of the tank is approximately 16.06 m³.