Question

The Candy Factory sells candy by the pound, charging $1.60 per pound for quantities up to and including 20 pounds. Above 20 pounds, the Candy Factory charges $1.30 per pound for the entire quantity, plus a quantity surcharge k. If x represents the number of pounds, the price function is p(x) = {1.60x for x lessthanorequalto 20, 1.30x + k for x > 20. Find k such that the price function p is continuous at x =20. Explain why it is preferable to have continuity at x =20. The price function p is continuous at x = 20 when k = (Type an integer or a decimal.)

Answer 1

Answer:

$k=6$

Step-by-step explanation:

We have been given that the Candy Factory sells candy by the pound, charging $1.60 per pound for quantities up to and including 20 pounds. Above 20 pounds, the Candy Factory charges $1.30 per pound for the entire quantity, plus a quantity surcharge k.

$\left \{ {{p(x)=1.6x, \text{ If }x\leq 20} \atop {p(x)=1.30x+x,\text{ if }x>20}} \right.$

For the price function be to continuous at $x=20$, the value of both functions at $x=20$ must be equal. Because a function is continuous if the right hand limit is equal to left hand limit.

$1.6(20)=1.3(20)+k$

Let us solve for k.

$32=26+k$

$32-26=26-26+k\\\\6=k$

Therefore, the price function p is continuous at  $x=20$, when k is equal to 6.