To solve this problem, first we need to establish a relationship between the semi vertical angle
θ of the cone, the radius
r of its base, and its altitude
h. The semi vertical angle within a cone relates the radius of the base to the altitude such that:
r=h‌tan(θ)We are given that the altitude
h=20‌cm is constant. The semi vertical angle
θ is, however, changing, and its rate of change is given by
‌=2∘ per second.
First, convert the rate from degrees to radians (since radians are a more natural measurement in calculus). Recall that
1∘=‌ radians, so:
‌=2∘ per second
=2×‌ radians per second
=‌ radians per second
Next, apply implicit differentiation to the relationship
r=20‌tan(θ) :
‌=20‌[tan(θ)]Using the chain rule,
‌[tan(θ)]= sec2(θ)‌.
‌=20 sec2(θ)‌=20 sec2(30∘)‌Simplifying this, using that
sec(30∘)=‌ (since
cos(30∘)=‌ and
sec(θ)=‌), we find:
sec2(30∘)=(‌)2=‌Therefore:
per second (since
π≈3.14)The radius of the base of the cone is therefore increasing at a rate of approximately
‌‌cm‌/‌ sec, which corresponds to:
Option C
‌‌cm‌/‌ sec