Suppose that y varies directly as the 3/2 power of x, and that y=192 when x= 16. find y when x=9.
y = k * x^(3/2)
192 = k * 16^(3/2)
192 = k * 4^3
192 = k * 64
k = 3
so
y = 3 x^(3/2)
if x = 9
y = 3 * 3^3
y = 3^4
y = 9*9
y = 81
ed
Well, well, well! Looks like we have a direct variation problem in our hands. Let's put on our mathematical clown shoes and solve it.
We're told that y varies directly as the 3/2 power of x. In other words, we can say y = kx^(3/2), where k is a constant.
Now, we're given that y = 192 when x = 16. So, we can plug those values into our equation:
192 = k(16)^(3/2)
To make things less "radical," let's rewrite that equation:
192 = k * (√16)^3
192 = k * 4^3
192 = k * 64
Now, time to solve for the constant k:
k = 192 / 64
k = 3
So, our equation becomes y = 3x^(3/2).
Now, we need to find y when x = 9:
y = 3(9)^(3/2)
And after a bit of clownish calculation:
y = 3 * (√9)^3
y = 3 * 3^3
y = 3 * 27
y = 81
There you have it! When x = 9, y equals 81. Voilà!
To find the value of y when x=9, we can use the concept of direct variation and the given information.
Let's start by writing the direct variation equation:
y = k * x^(3/2)
Here, k represents the constant of variation that relates y and x.
Now, let's substitute the values of y and x from the given information into the equation:
192 = k * 16^(3/2)
To simplify this equation, we need to evaluate 16^(3/2). The exponent 3/2 means taking the square root of 16 and then raising it to the power of 3.
16^(3/2) = (sqrt(16))^3 = 4^3 = 64
Now, we can rewrite the equation as:
192 = k * 64
To find the value of k, we divide both sides of the equation by 64:
k = 192 / 64 = 3
Now that we know the value of k, we can substitute it back into the direct variation equation:
y = 3 * x^(3/2)
Finally, we can find y when x=9:
y = 3 * 9^(3/2)
To evaluate 9^(3/2), we take the square root of 9 and then raise it to the power of 3.
9^(3/2) = (sqrt(9))^3 = 3^3 = 27
Substituting this into the equation, we get:
y = 3 * 27 = 81
Therefore, when x=9, y=81.