Page 49 - How To Solve Word Problems In Calculus

smaller, indicating that the projectile is returning to the
ground. The projectile returns to the ground after 16 sec.

Applications to Science and Engineering
EXAMPLE 9
The linear density of a rod is the rate of change of its mass with
respect to its length. A nonhomogeneous rod has a length of
9 feet and a total mass of 24 slugs. If the mass of a section of
the rod of length x (measured from its leftmost end) is propor-
tional to the square root of this length,

(a) Compute the average density of the rod.
(b) Determine the density function and compute the density
of the rod 4 ft from its leftmost end.

−
mass = k x √

O x 9

Solution
We let m(x) represent the mass of the section of the rod
of length x measured from its leftmost end. The description of
√
the problem tells us that m(x) = k x. Since the total mass of
the rod is 24 slugs, m(9) = 24.

m(9) = 24
√
k 9 = 24
3k = 24

k = 8

√
Thus m(x) = 8 x.
m(9) − m(0) 24 − 0 8
(a) The average density is = = slugs/ft
9 − 0 9 3
(b) We represent the density function by ρ(x). ρ(x) = m (x).

√
m(x) = 8 x = 8x 1/2

36