Question

the candy factory sells candy by the pound, charging $1.25 per pound for quantities up to and including 35 pounds. above 35 pounds, the candy factory charges $1.20 per pound for the entire quantity, plus a quantity surcharge k if x represents the number of pounds, the price function is as follows. p(x) = { 1.25x for x ≤ 35; 1.20x + k for x > 35 } find k such that the price function p is continuous at x = 35. the price function p is continuous at x = 35 when k = □ (type an integer or a decimal.)

Explanation:

Step1: Recall continuity condition

For a function to be continuous at $x = 35$, $\lim_{x
ightarrow35^{-}}p(x)=\lim_{x
ightarrow35^{+}}p(x)$.

Step2: Calculate left - hand and right - hand limits

Left - hand limit: $\lim_{x
ightarrow35^{-}}p(x)=1.25\times35 = 43.75$. Right - hand limit: $\lim_{x
ightarrow35^{+}}p(x)=1.20\times35 + k$.

Step3: Set limits equal

$1.20\times35 + k=43.75$.

Step4: Solve for k

$42 + k=43.75$, so $k = 43.75-42=1.75$.

Answer:

$1.75$