Question

A new fast-food firm predicts that the number of franchises for its products will grow at the rate dn dt = 6 t + 1 where t is the number of years, 0 ≤ t ≤ 15.

Answer 1

Answer:

The answer is "253"

Step-by-step explanation:

In the given- equation there is mistype error so, the correct equation and its solution can be defined as follows:

Given:

$\bold{\frac{dn}{dt} = 6\sqrt{t+1}}$

$\to dn= 6\sqrt{t+1} \ \ dt.....(a)$

integrate the above value:

$\to \int dn= \int 6\sqrt{t+1} \ \ dt \\\\\to n= \frac{(6\sqrt{t+1} )^{\frac{3}{2}}}{\frac{3}{2}}+c\\\\\to n= \frac{(12\sqrt{t+1} )^{\frac{3}{2}}}{3}+c$

When the value of n=1 then t=0

$\to 1= \frac{12(0+1)^{\frac{3}{2}}}{3}+c\\\\ \to 1= \frac{12(1)^{\frac{3}{2}}}{3}+c\\\\\to 1-\frac{12}{3}=c\\\\\to \frac{3-12}{3}=c\\\\\to \frac{-9}{3}=c\\\\\to c=-3$

so the value of  n is:

$\to n= \frac{(12\sqrt{t+1} )^{\frac{3}{2}}}{3}-3$

when we put the value t= 15 then,

$\to n= \frac{(12\sqrt{15+1} )^{\frac{3}{2}}}{3}-3\\\\\to n= \frac{(12\sqrt{16} )^{\frac{3}{2}}}{3}-3\\\\\to n= \frac{(12\times 64)}{3}-3\\\\\to n= (4\times 64)-3\\\\\to n= 256-3\\\\\to n= 253$