suppose that S varies directly as the 2/5 power of T, and that S=8 when T=32. Find S when T=243.
S = k(T)^(2/5)
when S = 8 , T = 32
8 = k (32)^(2/5)
8 = k(4)
k = 2
when T = 243
S = 2(243)^(2/5)
= 2(3^2) = 18
What does "k" stand for??
Well, solving this math problem requires a straightforward approach combined with a touch of clownishness. But don't worry, I got you covered!
To find the relationship between S and T, we know that S varies directly as the 2/5 power of T. It might sound a bit odd, but hey, math can be funny sometimes!
Let's set up the equation:
S ∝ T^(2/5)
Now, we need to find the constant of proportionality. We'll use the given information that S equals 8 when T equals 32.
8 = k * 32^(2/5)
Since we're clowns, let's simplify this equation by taking the 2/5 power of 32:
8 = k * (2^5)^(2/5)
8 = k * 2^(2)
8 = k * 4
Solving for k, we find that k = 2.
Now that we know the constant of proportionality, we can find S when T equals 243:
S = k * T^(2/5)
S = 2 * 243^(2/5)
S = 2 * 3^(2)
S = 2 * 9
S = 18
So, when T equals 243, S would be 18. Ta-da! Math can be funny, don't you think?
To solve the problem, we will use the given information and the concept of direct variation.
The problem states that S varies directly as the 2/5 power of T. This means that the relationship between S and T can be expressed as:
S = k * T^(2/5)
Where S is the dependent variable (varies), T is the independent variable (does not change), and k is the constant of variation.
We are given that S = 8 when T = 32. We can substitute these values into the equation to solve for k:
8 = k * (32)^(2/5)
To simplify the equation, we can evaluate (32)^(2/5):
(32)^(2/5) = (2^5)^(2/5) = 2^(5*2/5) = 2^2 = 4
Therefore, the equation becomes:
8 = k * 4
Simplifying, we find that k = 2.
Now that we know the value of k, we can substitute it back into the equation for S to solve for the value when T = 243:
S = 2 * 243^(2/5)
To evaluate 243^(2/5), we can raise 243 to the power of 1/5 and then square the result:
243^(1/5) = 3
(243^(1/5))^2 = 3^2 = 9
Thus, the equation becomes:
S = 2 * 9 = 18
Therefore, when T = 243, S = 18.