Question

use the given cost tables for the same product from two d then solve the system to determine when the cost of the p will be.

1. two online spice retailers sell paprika by the pound using

paprika (lb)	ispice	spice magic
2	$27.50	$36.50
3	$39.25	$46.75
4	$51	$57

Explanation:

Step1: Find the cost function for iSpice

Let \( x \) be the number of pounds of paprika. For iSpice, when \( x = 1 \), cost is \( \$15.75 \); when \( x = 2 \), cost is \( \$27.50 \). The slope (rate of change) \( m_{iS} = 27.50 - 15.75 = 11.75 \). Using point - slope form \( y - y_1=m(x - x_1) \) with \( (x_1,y_1)=(1,15.75) \), the cost function \( C_{iS}(x)=15.75 + 11.75(x - 1)=11.75x+4 \) (simplify: \( 15.75-11.75 = 4 \)).

Step2: Find the cost function for Spice Magic

For Spice Magic, when \( x = 1 \), cost is \( \$26.25 \); when \( x = 2 \), cost is \( \$36.50 \). The slope \( m_{SM}=36.50 - 26.25 = 10.25 \). Using point - slope form with \( (x_1,y_1)=(1,26.25) \), the cost function \( C_{SM}(x)=26.25+10.25(x - 1)=10.25x + 16 \) (simplify: \( 26.25 - 10.25=16 \)).

Step3: Set the cost functions equal to find the break - even point

Set \( C_{iS}(x)=C_{SM}(x) \), so \( 11.75x + 4=10.25x+16 \). Subtract \( 10.25x \) from both sides: \( 11.75x-10.25x + 4=16 \), which simplifies to \( 1.5x+4 = 16 \). Subtract 4 from both sides: \( 1.5x=16 - 4=12 \). Then divide both sides by 1.5: \( x=\frac{12}{1.5}=8 \).

Step4: Find the cost at \( x = 8 \)

Substitute \( x = 8 \) into \( C_{iS}(x) \): \( C_{iS}(8)=11.75\times8 + 4=94+4=\$98 \). We can also check with \( C_{SM}(x) \): \( C_{SM}(8)=10.25\times8+16 = 82 + 16=\$98 \).

Answer:

The cost of the paprika from both retailers will be equal when 8 pounds of paprika are purchased, and the cost will be \$98.