Question

1. multi-step jeds online music club allows him to download 25 songs per month for $14.99. additional songs cost $1.29 each. 47

a. write an inequality to represent this situation. let t be his monthly spending limit and m represent the total number of songs downloaded.
a ( 1.29m leq t + 10.01 )
b ( 1.29m leq t + 17.26 )
c ( 1.29t leq m + 10.01 )
d ( 1.29t leq m + 17.26 )
b. which of the following statements best describes the number of songs jed can download each month without going over his spending limit?
a jed will not go over his spending limit of $30 if he downloads 37 songs.
b jed will not go over his spending limit of $40 if he downloads 45 songs.
c jed can download a maximum of 36 songs if his budget is $30.
d jed can download a maximum of 36 songs if his budget is $40.
c. suppose that the music club changes its plan so that 50 songs can be downloaded a month. how will the inequality representing this inequality change?
a the coefficient of t will decrease.
b the coefficient of t will increase.
c the coefficient of m will increase.
d the constant will increase.

Explanation:

Step1: Analyze part a

The cost for \(m\) songs: The first 25 songs cost \(14.99\). If \(m\geq25\), the number of additional songs is \(m - 25\). The cost function \(C(m)=14.99+1.29(m - 25)=1.29m+14.99 - 1.29\times25=1.29m-17.26\). Since \(C(m)\leq t\), we can rewrite it as \(1.29m\leq t + 17.26\).

Step2: Analyze part b

For option A: If \(t = 30\) and \(m = 37\), \(C(37)=14.99+1.29\times(37 - 25)=14.99+1.29\times12=14.99 + 15.48=30.47>30\).
For option B: If \(t = 40\) and \(m = 45\), \(C(45)=14.99+1.29\times(45 - 25)=14.99+1.29\times20=14.99+25.8 = 40.79>40\).
For option C: If \(t = 30\), then \(1.29m-17.26\leq30\), \(1.29m\leq30 + 17.26=47.26\), \(m\leq\frac{47.26}{1.29}\approx36.64\). Since \(m\) is an integer, \(m = 36\).
For option D: If \(t = 40\), then \(1.29m-17.26\leq40\), \(1.29m\leq40+17.26 = 57.26\), \(m\leq\frac{57.26}{1.29}\approx44.4\).

Step3: Analyze part c

The original cost function was \(C(m)=14.99+1.29(m - 25)\) (when \(m\geq25\)). If 50 songs are included in the base - price, the new cost function is \(C(m)=14.99+1.29(m - 50)\) (when \(m\geq50\)). The coefficient of \(m\) remains \(1.29\), and the constant term \(14.99-1.29\times50=14.99 - 64.5=-49.51\) (more negative compared to the original constant \(14.99-1.29\times25=-17.26\)).

Answer:

a. B
b. C
c. D