Question

Sales of a​ product, under relatively stable market conditions but in the absence of promotional activities such as​ advertising, tend to decline at a constant yearly rate. This rate of sales decline varies considerably from product to​ product but seems to remain the same for any particular product. The sales decline can be expressed by the function defined by the next formula. S(t)=S0e−at In this formula​ S(t) is the rate of sales at time t measured in​ years, S0 is the rate of sales at time t=0 and a is the sales decay constant. If a=0.07 and S0=100,000​, find the number of years it will take for sales to fall to half the initial sales.

Answer 1

Q. Given: -at S(t)- So e s(t) is rate of sales at time t So is initial sale a is sales decay constant t is time in years a=0.07 So = 100,000 To find: time when sale is halved Sol? When the sale is halved the initial sales then Slt) - So 100,000 2 s(t) = 50,000 Se 500,000 n Putting all values in given eq -0.07 t 100,000 e 50,000 - -0.07t eno 2 ln ( 1 ) taking natural log both side en eo.070 -0.693 - -0.07 t xlne {: In oʻz bilmay {: line = 1 =t X -0.693 -0.07 t 9.902 years So the sale will be halved in 9.9 be halved in 9.9 years