To determine how much work must be done on a machine given that it has an efficiency of 80% and needs to produce 50,000 J of work, we can follow these steps:
1. Understand the concept of efficiency:
Efficiency ([tex]\( \eta \)[/tex]) is defined as the ratio of the useful work output to the total work input. It can be expressed as:
[tex]\[
\eta = \frac{\text{Work Output}}{\text{Work Input}}
\][/tex]
Given:
[tex]\[
\eta = 0.80
\][/tex]
[tex]\[
\text{Work Output} = 50,000 \, \text{J}
\][/tex]
2. Rearrange the efficiency formula to solve for Work Input:
[tex]\[
\eta = \frac{\text{Work Output}}{\text{Work Input}}
\][/tex]
Rearranging for Work Input:
[tex]\[
\text{Work Input} = \frac{\text{Work Output}}{\eta}
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
\text{Work Input} = \frac{50,000 \, \text{J}}{0.80}
\][/tex]
4. Calculate the Work Input:
[tex]\[
\text{Work Input} = 62,500 \, \text{J}
\][/tex]
So, the machine requires 62,500 J of work to be done on it to produce 50,000 J of useful work.
Therefore, the correct answer is [tex]$62,500 \, \text{J}$[/tex].
