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.)

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.

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

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

[tex]1.6(20)=1.3(20)+k[/tex]

Let us solve for k.

[tex]32=26+k[/tex]

[tex]32-26=26-26+k\\\\6=k[/tex]

Therefore, the price function p is continuous at [tex]x=20[/tex], when k is equal to 6.

chilljain33 chilljain33 · Answer 2 · 2022-01-07T08:11:39+01:00

The correct statement is that the real value of k is calculated as 6 given that the price function is continuous and the value of x is 20. This value will give optimum utilization of capital employed by the factory.

The calculation of k when x is either greater than the value of 20 and the similar value of k when the value of x is less than or equal to 20 is given as below.

Price function will help to determine whether the firm is achieving optimum utilization of its resources such as capital, labor, manpower and machines and equipment.

The price function of x is given as 20. The calculation of value of k in both the scenarios is as given below,

[tex]1.6(x)=1.3x +k[/tex]

We have derived the above formula after applying the information given given in the quoted statements. It is also given that the value of x is 20. applying x=20 in equation, we get,

[tex]1.6(20)=1.3(20)+k\\\\32= 26+k[/tex]

Continuing further we get the value of k as,

[tex]k = 32-26\\\\k=6[/tex]

So we know that the value of price function is constant at values x=20 and k=6

Hence, the correct statements is that the value of k is 6 when the price function is continuous at x=20 and the factory will benefit using this pricing method so as to achieve optimum utilization sources.

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