vitorpteixeira
Is cone P similar to cone Q?
(1) The surface area of P is 9 times the surface area of Q
(2) The volume of P is 9 times the volume of Q
HI..
with 100% answering incorrectly, this may help
thing to remember..
Two cones will be similar when the RADIUS and the HEIGHT are in SAME proportionso \(\frac{R_P}{R_Q}=\frac{H_P}{H_Q}=k\)
let's see the statements
(1) The surface area of P is 9 times the surface area of Q this means \(9*pi*R_P^2+pi*R_P*\sqrt{H_P^2+R_P^2}=pi*R_Q^2+pi*R_Q*\sqrt{H_Q^2+R_Q^2}\)..
so \(9*=pi*R_P^2+pi*R_P*\sqrt{H_P^2+R_P^2}/pi*R_Q^2+pi*R_Q*\sqrt{H_Q^2+R_Q^2}=(R_P^2+R_P*\sqrt{H_P^2+R_P^2})/(R_Q^2+R_Q*\sqrt{H_Q^2+R_Q^2})\)
two cases..
if similar, the ratio \(k = \sqrt{9}=3\)... Possible
or it can be various combination to get the ratio as 9...Possible
Insuff
(2) The volume of P is 9 times the volume of Q\(\frac{1}{3}*pi*R_P^2H_P=9*\frac{1}{3}*pi*R_Q^2H_Q............\frac{R_P^2H_P}{R_Q^2H_Q}=9\)..
two cases
1) if \(k^3=9\) .. similar possible
2) only \(R_P=3*R_Q\)... not similar.. possible
so insuff..
combined..different values of k in both cases so the cones are NOT similar
Sufficient
C