The heat experienced by a hiker at a campfire is proportional to the amount of wood on the fire and inversely proportional to the cube of his distance from the fire if the hiker is 18ft from the fire and someone doubles the amount of wood burning how far from the fire would he have to be so that he feels the same heat as before?(round your answer to one decimal place) in ft?
Let's denote the initial amount of wood on the fire as W1 and the initial distance from the fire as D1. We'll also call the initial heat experienced by the hiker as H1.
According to the given information, we have the following proportionalities:
Heat ∝ Wood
Heat ∝ 1/Distance^3
So we can write the equation as follows:
H1 ∝ W1 / D1^3
Now, let's say someone doubles the amount of wood burning, the new amount of wood on the fire is 2W1. Let's call the new distance from the fire as D2.
For the heat to remain the same, we can set up a new equation:
H1 = (2W1) / D2^3
We want to find D2, the new distance from the fire. To do so, we can rearrange the equation:
D2^3 = (2W1) / H1
D2 = (2W1 / H1)^(1/3)
Now, we substitute the given values:
H1 = 18ft (as given)
W1 = amount of wood on fire
D1 = 18ft (as given)
Since we don't know the actual amount of wood on the fire, we'll denote it as W1.
D2 = (2W1 / 18)^(1/3)
D2 = (2W1)^(1/3) / 18^(1/3)
D2 = (2)^(1/3) * W1^(1/3) / (18^(1/3))
D2 = (2)^(1/3) * W1^(1/3) / 2
D2 = (2)^(1/3) * (W1/2)^(1/3)
Thus, if the person doubles the amount of wood burning and wants to feel the same heat as before, they would have to be approximately (2)^(1/3) ≈ 1.26 times as far from the fire.
Therefore, if the initial distance from the fire was 18ft, the person would have to be approximately 1.26 * 18ft ≈ 22.7ft from the fire to feel the same heat as before.
To solve this problem, we can use the concept of the inverse square law. Let's denote the initial amount of wood as W1, the initial distance as D1, the final amount of wood as W2, and the final distance as D2.
According to the problem, the heat experienced is proportional to the amount of wood and inversely proportional to the cube of the distance. Mathematically, we can represent this as:
Heat1 ∝ W1 / (D1^3)
Heat2 ∝ W2 / (D2^3)
Since we want to find the final distance, D2, when the heat remains the same, we can set up the following equation:
Heat1 = Heat2
W1 / (D1^3) = W2 / (D2^3)
Given in the problem that the amount of wood doubles, we can express W2 in terms of W1:
W2 = 2W1
Substituting this into the equation, we have:
W1 / (D1^3) = (2W1) / (D2^3)
Canceling out W1, we have:
1 / (D1^3) = 2 / (D2^3)
To find D2, we need to isolate it:
(D2^3) = (D1^3) / 2
Taking the cube root of both sides, we get:
D2 = (D1 / 2)^(1/3)
Given that D1 = 18ft, let's substitute this value into the equation:
D2 = (18 / 2)^(1/3)
Simplifying:
D2 = 9^(1/3)
Using a calculator, we find:
D2 ≈ 2.08 ft (rounded to one decimal place)
Therefore, the hiker would have to be approximately 2.08 ft from the fire to feel the same heat as before.