Question

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Explanation:

Part (a)

Step 1: Find the height of water added

First, we need to find the height of the water added. The tank's total height is 60 cm? Wait, no, wait. Wait, the tank is 105 cm by 80 cm by 60 cm? Wait, no, the initial filling is 1/4 filled. Wait, the new water level is 25 cm from the top. Wait, let's re-express.

Wait, the tank dimensions: length \( l = 105 \) cm, width \( w = 80 \) cm, height \( h = 60 \) cm? Wait, no, maybe the height is 60 cm? Wait, the initial volume is \( \frac{1}{4} \) of the tank. Then, after adding water, the water level is 25 cm from the top. So the height of water after adding is \( 60 - 25 = 35 \) cm? Wait, no, maybe the tank's height is 60 cm? Wait, let's check.

Wait, initial volume: \( \frac{1}{4} \) of the tank. The tank's volume is \( 105 \times 80 \times 60 \) cubic cm. Wait, no, maybe the height is 60 cm. Wait, initial water height: \( \frac{1}{4} \times 60 = 15 \) cm. Then, after adding water, the height is \( 60 - 25 = 35 \) cm. So the height added is \( 35 - 15 = 20 \) cm? Wait, no, maybe I got the height wrong. Wait, the problem says "the new water level was 25 cm from the top". So the height of water is \( \text{total height} - 25 \). Wait, what's the total height? The tank's height is 60 cm? Wait, the tank is 105 cm (length) × 80 cm (width) × 60 cm (height). So total height \( H = 60 \) cm. So initial water height: \( \frac{1}{4} \times 60 = 15 \) cm. After adding, water height is \( 60 - 25 = 35 \) cm. So the height of water added is \( 35 - 15 = 20 \) cm.

Step 2: Calculate the volume of water added

The volume of water added is the volume of the rectangular prism with length 105 cm, width 80 cm, and height 20 cm.

Volume \( V = l \times w \times h = 105 \times 80 \times 20 \) cubic cm.

Calculate that: \( 105 \times 80 = 8400 \), then \( 8400 \times 20 = 168000 \) cubic cm.

Since 1 litre = 1000 cubic cm, convert to litres: \( \frac{168000}{1000} = 168 \) litres? Wait, that can't be right. Wait, maybe I messed up the height.

Wait, maybe the tank's height is 60 cm? Wait, let's re-express.

Wait, the tank's volume is \( 105 \times 80 \times 60 = 504000 \) cubic cm, which is \( 504000 \div 1000 = 504 \) litres (since 1 litre = 1000 cubic cm). Initial volume: \( \frac{1}{4} \times 504 = 126 \) litres.

After adding water, the water level is 25 cm from the top. So the height of water is \( 60 - 25 = 35 \) cm. So the volume of water after adding is \( 105 \times 80 \times 35 \) cubic cm.

Calculate that: \( 105 \times 80 = 8400 \), \( 8400 \times 35 = 294000 \) cubic cm = 294 litres.

So the water added is \( 294 - 126 = 168 \) litres. Wait, that matches the previous calculation. So that's the volume added.

Step 1: Find the total volume of the tank

The tank's volume is \( 105 \times 80 \times 60 \) cubic cm.

Calculate that: \( 105 \times 80 = 8400 \), \( 8400 \times 60 = 504000 \) cubic cm. Convert to litres: \( 504000 \div 1000 = 504 \) litres.

Step 2: Find the remaining volume to fill

The tank was initially \( \frac{1}{4} \) filled, so the remaining volume is \( 504 - \frac{1}{4} \times 504 = 504 - 126 = 378 \) litres? Wait, no. Wait, after adding water in part (a), the water is 294 litres. Wait, no, part (b) is: "If the capacity of a pail was 15 l, how many such pails of water were needed to fill the tank to its brim?"

Wait, the total volume of the tank is \( 105 \times 80 \times 60 = 504000 \) cubic cm = 504 litres.

After part (a), the water in the tank is 294 litres (from part a: 294 litres). Wait, no, part (b) is: after adding water to the level 25 cm from the top, how many pails are needed to fill to brim.

Wait, the volume from the current water level (35 cm height) to the top (60 cm height) is the volume to fill. The height to fill is \( 60 - 35 = 25 \) cm.

So the volume to fill is \( 105 \times 80 \times 25 \) cubic cm.

Calculate that: \( 105 \times 80 = 8400 \), \( 8400 \times 25 = 210000 \) cubic cm = 210 litres.

Then, number of pails: \( \frac{210}{15} = 14 \) pails.

Wait, let's check again. Total tank volume: \( 105 \times 80 \times 60 = 504000 \) cm³ = 504 L.

Current water volume (after part a): 294 L (from part a: 294 L). So remaining volume: \( 504 - 294 = 210 \) L.

Number of pails: \( 210 \div 15 = 14 \).

Answer:

(a):
\( \boxed{168} \) litres

Part (b)