T is directly proportional to r². It is given that T=9 for a particular value of r. Find the value of T when this value of r is doubled.

Mathematically, if T is directly proportional to r squared, we write that as:
T∝r^2
To get from a proportional expression (∝) to an equation (=), we have to add a multiplication factor to r^2 to account for the how fast T changes with respect to r^2, so:
T=k*(r)^2 (1)
From here, we can plug in the numbers we know, which leads to:
9=k*(r)^2 (2)
And understanding that r doubling is the same as 2*r,
T=k*(2*r)^2 (3)
Using the distributive property we get:
T=k*(2)^2*(r)^2 (4)
Which simplifies to:
T=4*k*(r)^2 (5)
Now, if we look to work we did up above in equation (2), we see that:
k*(r)^2=9 (6)
So, now we substitute 9 into equation (5) and get:
T=4*9=28,
So, T=28 when r is doubled.