Answer:
If y is inversely proportional to the cube root of x, we can write the equation:
y = k / (x^(1/3))
where k is a constant of proportionality. To find k, we can use the given information that x=64 when y=12.75:
12.75 = k / (64^(1/3))
Simplifying:
12.75 = k / 4
k = 51
So the equation connecting x and y is:
y = 51 / (x^(1/3))
To find the value of x when y=3, we can plug in y=3 into the equation we just found and solve for x:
3 = 51 / (x^(1/3))
Simplifying:
x^(1/3) = 51 / 3
x^(1/3) = 17
Cubing both sides:
x = 17^3
x = 4913
So when y=3, x=4913.
If we divide x by 125, we can write the new value of x as x/125. To find the new value of y, we can plug this into the equation we found in part 1:
y = 51 / ((x/125)^(1/3))
Simplifying:
y = 51 / ((1/125)^(1/3)) * (x^(1/3))
y = 51 / 5 * (x^(1/3))
So the new value of y is:
y' = 51 / 5 * ((x/125)^(1/3))
To find the change in y, we can subtract the original value of y from the new value:
Δy = y' - y
Δy = 51 / 5 * ((x/125)^(1/3)) - 51 / (x^(1/3))
Simplifying:
Δy = 51 / 5 * ((x/125)^(1/3)) - 51 * (x^(-1/3))
Note that we can simplify this expression further by finding a common denominator, but this is the final answer.
Explanation: