14. Tap A can fill a bucket in 4 hours, and a hole at the bottom can empty it in 12 hours.

a) After how much time will the bucket be filled if both the tap and the hole are opened simultaneously?

b) How many liters of water are wasted in 7 hours if the capacity of the bucket is 84 liters?



Answer :

Certainly! Let's break down the problem step-by-step.

### Time to Fill the Bucket
1. Filling Rate of the Tap:
- Tap A can fill the entire bucket in 4 hours.
- Therefore, the rate at which the tap fills the bucket is [tex]\( \frac{1}{4} \)[/tex] of the bucket per hour.

2. Emptying Rate of the Hole:
- The hole at the bottom can empty the entire bucket in 12 hours.
- Therefore, the rate at which the hole empties the bucket is [tex]\( \frac{1}{12} \)[/tex] of the bucket per hour.

3. Net Filling Rate:
- When both the tap and the hole are operating at the same time, the net filling rate is calculated as follows:
[tex]\[ \text{Net Filling Rate} = \frac{1}{4} - \frac{1}{12} \][/tex]

Let's simplify this:
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
[tex]\[ \text{Net Filling Rate} = \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \][/tex]

So, the net rate of the bucket being filled is [tex]\( \frac{1}{6} \)[/tex] of the bucket per hour.

4. Time to Fill the Bucket:
- To find out how long it will take to fill the bucket, we take the reciprocal of the net filling rate:
[tex]\[ \text{Time to Fill} = \frac{1}{\frac{1}{6}} = 6 \text{ hours} \][/tex]

### Water Wasted in 7 Hours
1. Net Emptying Rate:
- If we consider the negative effect (wasting water), we look at the rate at which water is wasted when the hole is emptying faster than the tap fills.
- The net emptying rate can be formulated as:
[tex]\[ \text{Net Emptying Rate} = \frac{1}{12} - \frac{1}{4} \][/tex]

Remember,
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
So,
[tex]\[ \text{Net Emptying Rate} = \frac{1}{12} - \frac{3}{12} = -\frac{2}{12} = -\frac{1}{6} \][/tex]

Here, the negative sign indicates that water is being wasted.

2. Volume of Water Wasted:
- With a net emptying rate of [tex]\( -\frac{1}{6} \)[/tex] per hour, and considering a bucket capacity of 84 L, the volume of water wasted over 7 hours is calculated as follows:
[tex]\[ \text{Water Wasted} = \left(-\frac{1}{6}\right) \times 7 \times 84 \][/tex]
[tex]\[ \text{Water Wasted} = -\frac{7 \times 84}{6} = -\frac{588}{6} = -98 \text{ liters} \][/tex]

### Final Answers
- The bucket will be filled in 6.0 hours.
- The amount of water wasted in 7 hours is -98.0 liters. The negative sign indicates that this volume is lost.