The density of a rod 9 meters long is $\sqrt { x }$ kg/m at a distance of x meters from one end of the rod. Find the average density of the rod.
A)6
B) $\frac { 4 } { 3 }$
C)3
D)1
E)2
F) $\frac { 16 } { 3 }$
G)4
H) $\frac { 8 } { 3 }$

Question

The density of a rod 9 meters long is $\sqrt { x }$ kg/m at a distance of x meters from one end of the rod. Find the average density of the rod.
A)6
B) $\frac { 4 } { 3 }$ 
C)3
D)1
E)2
F) $\frac { 16 } { 3 }$ 
G)4
H) $\frac { 8 } { 3 }$

Quizplus · Accepted Answer

The average density of the rod can be found by integrating the density function over the length of the rod and then dividing by the length of the rod. The density function is given as $\sqrt{x}$ kg/m. To find the total mass, integrate $\sqrt{x}$ from 0 to 9:$$\int_{0}^{9} \sqrt{x} \, dx = \int_{0}^{9} x^{1/2} \, dx = \frac{2}{3}x^{3/2} \Big|_{0}^{9} = \frac{2}{3}(9^{3/2} - 0) = \frac{2}{3} \cdot 27 = 18 \, \text{kg}$$The length of the rod is 9 meters, so the average density is the total mass divided by the total length:$$\text{Average density} = \frac{18 \, \text{kg}}{9 \, \text{m}} = 2 \, \text{kg/m}$$Therefore, the correct answer is 2 kg/m, which corresponds to choice E.

The Density of a Rod 9 Meters Long Is x\sqrt { x }x​

The Density of a Rod 9 Meters Long Is $\sqrt { x }$