For the first problem, we are given that y varies directly with the square of R. This can be represented by the equation y = kR^2, where k is the constant of variation.
To find the value of k, we can substitute the given values of y and R into the equation. We have y = 128 and R = 8. Substituting these values, we get:
128 = k * 8^2
128 = k * 64
Dividing both sides of the equation by 64, we find that k = 2.
Now that we have the value of k, we can use it to find y when R = 5. Substituting R = 5 and k = 2 into the equation, we get:
y = 2 * 5^2
y = 2 * 25
y = 50
So, when R = 5, y = 50.
For the second problem, we are given that x varies directly with y. This can be represented by the equation x = ky, where k is the constant of variation.
To find the value of k, we can substitute the given values of x and y into the equation. We have x = 32 and y = 8. Substituting these values, we get:
32 = k * 8
Dividing both sides of the equation by 8, we find that k = 4.
Now that we have the value of k, we can use it to find x when y = 10. Substituting y = 10 and k = 4 into the equation, we get:
x = 4 * 10
x = 40
So, when y = 10, x = 40.