A fish pond has a shape of a triangle whose sides are 15 m, 10 m and 8 m long. If the depth of the pond is 4 m, how many liters of water are needed to fill the pond?

Steps to Solve

1. Calculate the semi-perimeter $s$ of the triangular base

The semi-perimeter is half the sum of the lengths of the sides of the triangle. Given the sides $a = 15$ m, $b = 10$ m, and $c = 8$ m, we have:

$s = \frac{a + b + c}{2} = \frac{15 + 10 + 8}{2} = \frac{33}{2} = 16.5$ m

2. Calculate the area of the triangular base using Heron's formula

Heron's formula states that the area $A$ of a triangle with sides $a$, $b$, and $c$ and semi-perimeter $s$ is:

$A = \sqrt{s(s-a)(s-b)(s-c)}$

Plugging in the values, we get:

$A = \sqrt{16.5(16.5-15)(16.5-10)(16.5-8)} = \sqrt{16.5(1.5)(6.5)(8.5)} = \sqrt{1369.6875} \approx 37.01$ m$^2$

3. Calculate the volume of the triangular prism The volume $V$ of a triangular prism is given by the area of the triangular base $A$ multiplied by the depth $d$ (which is the height of the prism):

$V = A \times d$

$V = 37.01 \times 4 \approx 148.04$ m$^3$

4. Convert the volume from cubic meters to litres

Since 1 m$^3$ = 1000 litres, we multiply the volume in cubic meters by 1000 to get the volume in litres:

$V_{\text{litres}} = V_{\text{m}^3} \times 1000$

$V_{\text{litres}} = 148.04 \times 1000 = 148040$ litres