Full solution

Q. for which the line

y+2 x=7

is a tangent to the curve. week after the col the amount obtained in the preceding week. It is given that in the fir

\newline

(i) Find the company obtained

8000 \mathrm{~kg}

of salt.

\newline

(ii) Find the total amount of salt obtained in the first

12

weeks after the change.

Understand Initial Amount of Salt: Step $1$ : Understand the initial amount of salt obtained. $\newline$ The company started with $8000\,\text{kg}$ of salt in the first week.
Calculate Salt Obtained in Second Week: Step $2$ : Calculate the amount of salt obtained in the second week. $\newline$ The amount of salt obtained each week increases by the amount obtained in the preceding week. Therefore, in the second week, the company obtained $8000\,\text{kg}$ (from the first week) + $8000\,\text{kg}$ = $16000\,\text{kg}$ .
Calculate Total Salt Obtained in $12$ Weeks: Step $3$ : Calculate the total amount of salt obtained in the first $12$ weeks. $\newline$ Using the pattern from the previous steps, the amount doubles each week. So, the sequence of salt obtained over $12$ weeks forms a geometric series: $\newline$ Week $1$ : $8000\,\text{kg}$ $\newline$ Week $2$ : $16000\,\text{kg}$ $\newline$ Week $3$ : $32000\,\text{kg}$ , and so on. $\newline$ The sum of a geometric series $S_n = \frac{a(r^n - 1)}{(r - 1)}$ , where: $\newline$ a = first term = $8000\,\text{kg}$ , $\newline$ r = common ratio = $2$ , $\newline$ n = number of terms = $12$ . $\newline$ $S_{12} = \frac{8000(2^{12} - 1)}{(2 - 1)}$
Perform Calculation for Sum: Step $4$ : Perform the calculation for the sum of the geometric series. $S_{12} = 8000(4095) = 32760000 \, \text{kg}.$