If r = square root 3V/pi*h , by what number must we multiply V in order to multiply r by 9 ?
a. 3
b. 9/2
c. 9
d. 18
e. 81
please answer and explain
r^2 = k V
k = r^2/V
if you multiply r by 9, then V by 81
k = (9r)^2/81V = r^2/V
so e
To find the number by which we must multiply V to multiply r by 9, we can start by rearranging the equation:
r = √(3V/π * h)
Let's isolate V:
r = √(3V * h / π)
r² = 3V * h / π
r² * π / 3 = V * h
Now, we can calculate the ratio between the new r' and the original r:
r' = 9 * r
r'² * π / 3 = (9 * r) * h
r'² * π / 3 = 9 * (r * h)
Since r'² = (9 * r)² = 81 * r², let's substitute that in:
81 * r² * π / 3 = 9 * (r * h)
Simplifying further:
27 * r² * π = 9 * r * h
3 * r² * π = r * h
Now, since we are looking for the factor by which V should be multiplied, we can compare the terms involving V in the original equation (r = √(3V/π * h)) and the rearranged equation above:
Original equation: 3V/π = r²/h
Rearranged equation: 3 * r² * π = r * h
Comparing the terms, we see that 3V/π in the original equation is equal to r/h in the rearranged equation. Therefore, the number by which we must multiply V to multiply r by 9 is equal to r/h.
Therefore, the correct answer is (b) 9/2.