A one-parameter family of power–law curves is defined by
MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
A one-parameter family of power–law curves is defined by
y=kx3/2,0≤x≤4,
where k>0 is a constant. The total arc length of the curve on the stated interval is known to be 50 units.
(a) Derive an explicit expression for the arc length L(k) of the curve in terms of the parameter k.
(b) Using your result from part (a), determine the value of k (to three decimal places) for which the arc length equals 50.
(c) Verify that your value of k has the correct units by clearly identifying the units of every term in the arc–length expression and explaining how dimensional consistency is maintained.