Two cylinders P and Q are made of copper P
The height of P is twice the height of Q. The diameter of P is half the diameter of
Which statement is comrect?
A The density of cylinder Pis four times that of cylinder o
The density of cylinder Pis twice that of cylinder
The density of cylinder P is equal to that of cylinder Q
The density of cylinder P is half that of cylinder o
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D

Let H and D denote the height and diameter, respectively, of cylinder P, and h and d the height and diameter of cylinder Q.

Then the volumes of P and D, denoted V and v, respectively, are

V = π (D / 2)² H = π D ² H / 4

v = π (d / 2)² h = π d ² h / 4

The height of P is twice the height of Q, so H = 2h.

The diameter of P is half the diameter of Q, so D = d / 2.

Substitute these into equation for the volume of cylinder P:

V = π (d / 2)² (2h) / 4

V = π (d ² / 4) (2h) / 4

V = π d ² h / 8

V = 1/2 • π d ² h / 4

V = v / 2

That is, cylinder P has half the volume of cylinder Q.

Recall that density is equal to mass per unit of volume. So R and ρ, the respective densities of cylinders P and Q, are

R = m / V = m / (v / 2) = 2 m / v

ρ = m / v

which means cylinder P has twice the density of cylinder Q (assuming both cylinders have the same mass m).