(1 point)\nthe copy shoppe charges 0.1 dollars per copy for quantites up to and included 108 copies. for…

(1 point)\nthe copy shoppe charges 0.1 dollars per copy for quantites up to and included 108 copies. for quantities above 108, the charge is 0.08 dollars per copy plus an unknown constant k. if x represents the number of copies, the price function is:\np(x)=\\begin{cases}0.1x&x \\leq 108 \\\\ 0.08x + k&x>108\\end{cases}\nfind the value of k that makes the price function continuous for all values of x.\nthe price function is continuous if k = \nround to three decimal places if needed.

(1 point)\nthe copy shoppe charges 0.1 dollars per copy for quantites up to and included 108 copies. for quantities above 108, the charge is 0.08 dollars per copy plus an unknown constant k. if x represents the number of copies, the price function is:\np(x)=\\begin{cases}0.1x&x \\leq 108 \\\\ 0.08x + k&x>108\\end{cases}\nfind the value of k that makes the price function continuous for all values of x.\nthe price function is continuous if k = \nround to three decimal places if needed.

Answer

Explanation:

Step1: Recall continuity condition

For a function to be continuous at $x = 108$, $\lim_{x\rightarrow108^{-}}p(x)=\lim_{x\rightarrow108^{+}}p(x)$.

Step2: Calculate left - hand limit

$\lim_{x\rightarrow108^{-}}p(x)=0.1\times108 = 10.8$.

Step3: Calculate right - hand limit

$\lim_{x\rightarrow108^{+}}p(x)=0.08\times108 + k=8.64 + k$.

Step4: Set left and right limits equal

Set $10.8=8.64 + k$.

Step5: Solve for k

$k=10.8 - 8.64=2.16$.

Answer:

$2.160$