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.
