Exponential and Logarithmic functions

Sales and Advertising: The sales S (in thousands of units) of a product after x hundred dollars is spent on advertising is S=10(1-e^kx). When $500 is spent on advertising, 2500 units are sold.

(a) Complete the model by solving for K.
(b) Estimate the number of units that will be sold if advertising expenditures are raised to $700.

(a) Complete the model by solving for K.
S=10(1-e^kx).

Sales= S= 2500 =2.5 in thousands
Advertising= x= 500 =5 (in hundreds)

Therefore 2.5=10 * (1-e^5k)

or (1-e^5k)=2.5/10=0.25
or (1-0.25)=e^5k
or ln 0.75=5k
or k=ln 0.75/5= -0.057536

(b) Estimate the number of units that will be sold if advertising expenditures are raised to $700.

S=10(1-e^kx)=10(1-e^(-0.057536x))

x= 700 =7 in hundreds

S=10(1-e^(-0.057536*7))= 3.315221387 (in thousands)

= 3315 units

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