Question

(i) At what rate is the area of a rectangle changing if its length is 15 𝑓𝑡 and increasing at 3𝑓𝑡/𝑠𝑒𝑐 while its width is 6𝑓𝑡 and increasing at 2𝑓𝑡/𝑠𝑒𝑐? (ii) The pressure, 𝑝, and the volume, 𝑉, are connected by the equation 𝑝𝑉 1.4 = 𝐾. Determine the approximate percentage error in 𝑘 when the pressure is increased by 4% and the volume is decreased by 1.5%.

Answer 1

(i) Length, l is 15 ft and dl/dt = 3

Width, w is 6 ft and dw/dt = 2

Area, A = l x w

dA/dt = ldw/dt + wdl/dt

= 15 x 2 + 6 x 3

= 48 ft²/s

 

Area is increasing at the rate of 48 ft^2/s.

 

(ii) pv^1.4 = k

dp/p = 4%, dv/v = -1.5%

Taking log in initial equation,

logp + 1.4logv = logk

dp/p + 1.4dv/v = dk/k

dk/k = 4 + 1.4(-1.5)

= 1.9%