Suppose that S varies directly as the 4/5 power of T, and that S = 64 when T = 32. Find S when T = 243
s = k t^ 0.8
64 = k (32)^0.8
k = 64 / 32^0.8
log base 10 of k = 64 - 0.8 log 32
log k = 1.806 - 0.8 * 1.505
log k = 1.806 - 1.204 = .602
so
k = 4
s = 4 t^ 0.8
s = 4 (243)^0.8
= 4 * 81 = 324
To find the value of S when T = 243, we can use the direct variation equation:
S ∝ T^(4/5)
First, use the given values to determine the constant of variation.
We are given that S = 64 when T = 32:
64 ∝ 32^(4/5)
Now, raise both sides to the power of 5/4 to solve for the constant of variation:
(64)^(5/4) ∝ (32^(4/5))^(5/4)
1024 ∝ 32
Therefore, the constant of variation is 1024/32 = 32.
Now, substitute the given value T = 243 into the equation:
S ∝ T^(4/5)
S = 32 × 243^(4/5)
Calculate the value of 243^(4/5) using a calculator:
S = 32 × (243^(4/5)) ≈ 32 × 9.741 ≈ 311.712
Therefore, S is approximately equal to 311.712 when T = 243.
To find the value of S when T = 243, we can use the concept of direct variation, which states that when two variables are directly proportional, their ratio remains constant. In this case, S varies directly with the 4/5 power of T, which can be written as:
S = k * T^(4/5)
To solve for the constant of variation (k), we can substitute the given values:
64 = k * 32^(4/5)
First, we need to simplify the equation:
32^(4/5) = (2^5)^(4/5) = 2^4 = 16
Now, we can solve for k:
64 = k * 16
Divide both sides by 16:
64/16 = k
k = 4
Now that we have the value of k, we can substitute it into the original equation to find S when T = 243:
S = 4 * 243^(4/5)
To simplify this expression, we can evaluate 243^(4/5):
243^(4/5) = (3^5)^(4/5) = 3^4 = 81
Now, we can substitute this value into the equation:
S = 4 * 81
S = 324
Therefore, when T = 243, S is equal to 324.