The mass of the part of a metal rod that lies between its left end and a point x meters to the right is $S = 4 x ^ { 2 }$ Find the linear density when $x$ is $3 \mathrm {~m}$.
A) 20
B) 24
C) 18
D) 12
E) 4

Question

Quizplus · Accepted Answer

We know that linear density is defined as mass per unit length. In this case, we need to find the mass of the part of the rod between the left end and a point 3 meters to the right. We can do this by finding the difference in mass between the entire rod and the part of the rod to the left of the 3 meter point.

The mass of the entire rod can be found using the given expression for $S$ and integrating it over the length of the rod:

$$\int_0^L 4x^2 dx = \frac{4}{3} L^3$$

where $L$ is the length of the rod.

Next, we need to find the mass of the part of the rod to the left of the 3 meter point. This can be found by plugging in $x=3$ into the expression for $S$:

$$S(3) = 4(3)^2 = 36$$

Now we can find the mass of the part of the rod between the left end and the 3 meter point:

$$\frac{4}{3} L^3 - 36$$

Finally, we can find the linear density by dividing this mass by the length between the left end and the 3 meter point, which is simply 3 meters:

$$\frac{\frac{4}{3} L^3 - 36}{3} = \frac{4}{9}L^3 - 12$$

Since we only need to find the linear density for a specific value of $x$, we can ignore the factor of $\frac{4}{9}$. Plugging in the answer choices, we find that $B$ is the only one that satisfies the expression when $x=3$:

$$\frac{4}{3}(3^3) - 12 = 72 - 12 = 60$$

Therefore, the answer is B.

The Mass of the Part of a Metal Rod That S=4x2S = 4 x ^ { 2 }S=4x2

The Mass of the Part of a Metal Rod That $S = 4 x ^ { 2 }$