Sure, let's solve this step-by-step:
### Part (a)
First, we need to understand the relationship between power ([tex]\(P\)[/tex]) and time ([tex]\(T\)[/tex]). The problem states that "the power required to complete a task varies inversely to the time it takes to complete the task."
This can be expressed mathematically as:
[tex]\[ P \times T = k \][/tex]
where [tex]\(k\)[/tex] is a constant.
Given that it takes 20 watts to roll a stone across a field in 47 seconds, we can use these values to find the constant [tex]\(k\)[/tex]:
[tex]\[ P = 20 \text{ watts} \][/tex]
[tex]\[ T = 47 \text{ seconds} \][/tex]
Substituting these values into the equation:
[tex]\[ 20 \times 47 = k \][/tex]
[tex]\[ k = 940 \][/tex]
So, the relationship between power and time is given by:
[tex]\[ P \times T = 940 \][/tex]
### Part (b)
Now, we need to find how many watts of power would be required to roll the stone across the field in 10 seconds. Let's denote the new power requirement as [tex]\(P_{\text{new}}\)[/tex] and the new time as [tex]\(T_{\text{new}}\)[/tex].
Given:
[tex]\[ T_{\text{new}} = 10 \text{ seconds} \][/tex]
We use the relationship [tex]\( P \times T = 940 \)[/tex] to find [tex]\( P_{\text{new}} \)[/tex]:
[tex]\[ P_{\text{new}} \times 10 = 940 \][/tex]
Solving for [tex]\( P_{\text{new}} \)[/tex]:
[tex]\[ P_{\text{new}} = \frac{940}{10} \][/tex]
[tex]\[ P_{\text{new}} = 94 \text{ watts} \][/tex]
Rounding to three decimal places (although in this case it remains the same):
[tex]\[ P_{\text{new}} = 94.000 \text{ watts} \][/tex]
Thus, it would take 94.000 watts of power to roll the stone across the field in 10 seconds.
