Question

1. annette has a rectangular - shaped cistern in her basement to contain water for use in her house. the cistern has a base of 1.5 m by 1 m, and the water is currently at a level of 15 cm. annette wants to know if she has enough water to fill her cylindrical hot water tank before filling the cistern again. the hot water tank has a capacity of 140 l. the tank has a radius of 25 cm and a height of 110 cm.

a. draw diagrams of the cistern and the hot water tank with dimensions.
b. find the volume of water in the cistern to the nearest cm³.
c. find the height the water will reach in the hot water tank to the nearest cm.
d. will the water fit into the hot water tank? will she have enough water to fill it?

Explanation:

Step1: Convert cistern base dimensions to cm

Since 1 m = 100 cm, the base of the cistern has dimensions 1.5 m = 150 cm and 1 m = 100 cm.

Step2: Calculate volume of water in cistern

The volume \(V\) of a rectangular - prism (water in cistern) is \(V = l\times w\times h\), where \(l = 150\) cm, \(w = 100\) cm and \(h = 15\) cm. So \(V=150\times100\times15=225000\) \(cm^{3}\).

Step3: Calculate volume of hot - water tank in \(cm^{3}\)

The volume \(V\) of a cylinder is \(V=\pi r^{2}h\), with \(r = 25\) cm and \(h = 110\) cm. So \(V=\pi\times(25)^{2}\times110=\pi\times625\times110 = 68750\pi\approx 68750\times 3.14 = 215875\) \(cm^{3}\). Also, since 1 L = 1000 \(cm^{3}\), the capacity of the tank is \(140\times1000 = 140000\) \(cm^{3}\).

Step4: Find height of water in hot - water tank if filled with cistern water

We know the volume of water from the cistern \(V = 225000\) \(cm^{3}\) and the base - area of the cylinder \(A=\pi r^{2}=\pi\times(25)^{2}=625\pi\) \(cm^{2}\). Using the formula \(V = A\times h\), we get \(h=\frac{V}{A}=\frac{225000}{625\pi}=\frac{225000}{625\times3.14}\approx114.65\) cm \(\approx115\) cm.

Step5: Answer part d

Since the volume of water in the cistern is \(225000\) \(cm^{3}\) and the capacity of the hot - water tank is \(140000\) \(cm^{3}\), the water will not fit entirely into the hot - water tank and she does not have enough water to fill it.

Answer:

a. (Diagrams are not provided in text - based response. For the cistern, draw a rectangle with length 150 cm and width 100 cm as the base and height 15 cm. For the hot - water tank, draw a cylinder with radius 25 cm and height 110 cm).
b. \(225000\) \(cm^{3}\)
c. \(115\) cm
d. The water will not fit into the hot - water tank. She does not have enough water to fill it.