Question

The members of a marching band form a rectangle with initial length L0=40 feet and initial width W0=20 feet. As their song begins, they begin marching so that the length of the rectangle increases at a rate of 3 feet per second, and the width of the rectangle increases at a rate of 2 feet/second.
(a) Write an equation for area A of the rectangle of the marching band in terms of length L and width W.
(b) Use implicit differentiation to find an expression for da/dt in terms of W, L, dw/dt, and dl/dt.
(c) Write expressions for L(t) and W(t) in feet, in terms of time t in seconds.
(d) How fast is the area of the rectangle increasing t=5 seconds after their song begins? Give units.

Answer 1

Lo= 40 feet Wo =20 feet di = 3 feet /sec at dw = 2 feet/sec dit de 3 o.s.t. t. Integrating both side Sald = ßdt = 34+ c to, L = 40 40 - c IL = 3t+ 40 (1) dw=2 & an = fa de w = 2t to at o W 20 20 c w = 2t + 20

a Area of Rectangle A = Length x width Area = Low na ww where we retro where we att 26 2 = 3t+40 A = WL alifferentiate with respect to t A = (NL at at dA & wdl & L dw - 3) (using product Rule] olt (C) frow eau (1) L(t) = 3 + + 40 feet/sce. from egn (2) wit- 2t +20 feet/sec

Call Rate of change of Area at t-5 sec. da - wdl & I du at from egu (3) t=5 w=2 (5) +20 - 30 feet see 1=305) + 40 = 55 feet/sec da = (36) (3) + (55) (2) + A = 90+11o A = 200 feet/sec at Area is changing at the rate of 200 feetrec
(a) evaluated expression for area of rectangle

(b) evaluated the expression for dA/dt in terms of W, L, dW/dt , dL/dt

(c) evaluated expression for L(t) and W(t)

(d) calculated the rate of change of Area at t=5 sec

which is 200 feet²/sec