Certainly! Let's solve this step-by-step.
### Part (a)
To relate the power needed to roll the stone across the field to the time it would take, we start by recognizing that power (P) varies inversely to time (t). This means that the product of power and time is a constant (k).
We are given:
- Power initial, [tex]\(P_{i} = 30\)[/tex] watts
- Time initial, [tex]\(t_{i} = 45\)[/tex] seconds
The equation for inverse variation is:
[tex]\[ P \times t = k \][/tex]
Using the given values to find the constant [tex]\( k \)[/tex]:
[tex]\[ k = P_{i} \times t_{i} \][/tex]
[tex]\[ k = 30 \text{ W} \times 45 \text{ s} = 1350 \][/tex]
Therefore, the equation that relates the power (P) needed to the time (t) is:
[tex]\[ P \times t = 1350 \][/tex]
### Part (b)
Now, we need to find the power required ([tex]\(P_{n}\)[/tex]) to roll the stone across the same field in 33 seconds ([tex]\(t_{n} = 33\)[/tex] seconds).
Using the equation:
[tex]\[ P_{n} \times t_{n} = k \][/tex]
We substitute the known values:
[tex]\[ P_{n} \times 33 = 1350 \][/tex]
Solving for [tex]\(P_{n}\)[/tex]:
[tex]\[ P_{n} = \frac{1350}{33} \][/tex]
[tex]\[ P_{n} \approx 40.90909090909091 \][/tex]
Rounding this to 3 decimal places:
[tex]\[ P_{n} \approx 40.909 \][/tex]
Therefore, the power needed to roll the stone across the field in 33 seconds would be approximately 40.909 watts.
