Question

math 95 14 worksheet

1. you know you spent $600 on food and gas together last month. your credit card gives you 1% back on food and 2% back on gas, and you know you received $7.15 cash - back last month as a result. however, you cant remember exactly how much you spent on food and gas individually, and youre in a raging debate with your spouse. your internets down so you cant pull open your credit - card statements, and while youd usually just wait for internet to return, you seek the glory of figuring it out before your spouse. how much did you spend on food, and how much did you spend on gas?
2. you are fed up with your cell phone plan and are thinking of switching carriers. carrier a is offering a $35 per month plan with 3 gb (gigabytes) of data and then a charge of $8 per additional gb of data after that. similarly, carrier b is offering a $55 per month plan with 5 gb of data and then a charge of $4 per additional gb of data after that.

a. in a given month, after how much data use would the two plans cost the same?
b. if you historically use an average of 6 gb of data, which plan should you choose?
c. which plan is cheaper in the long run?

Explanation:

10.

Step1: Set up equations

Let $x$ be food cost, $y$ be gas cost. $x + y=600$, $0.01x + 0.02y=7.15$.

Step2: Substitute

Substitute $x = 600 - y$ into $0.01x + 0.02y=7.15$.

Step3: Solve for $y$

Simplify and solve $6-0.01y + 0.02y=7.15$ for $y$.

Step4: Solve for $x$

Substitute $y$ value into $x + y=600$ to find $x$.

11.
a.

Step1: Define cost functions

Define $C_A$ and $C_B$ based on data usage $d$.

Step2: Analyze cases

Analyze different cases of $d$ to find when $C_A = C_B$.

Step3: Solve for $d$

Solve $11 + 8d=35 + 4d$ for $d$ in the relevant case.

b.

Step1: Calculate costs

Calculate $C_A$ and $C_B$ when $d = 6$.

Step2: Make a decision

Compare costs to decide which plan to choose.

c.

Step1: Find cost difference

Find $C_A - C_B$ as a function of $d$.

Step2: Analyze long - run

Analyze the sign of $C_A - C_B$ for different $d$ values to determine the cheaper plan in the long - run.

Answer:

10.
Let $x$ be the amount spent on food and $y$ be the amount spent on gas.
We know two - equations:
$x + y=600$ (total amount spent on food and gas)
$0.01x + 0.02y=7.15$ (total cash - back)
From the first equation $x = 600 - y$.
Substitute $x = 600 - y$ into the second equation:
$0.01(600 - y)+0.02y=7.15$
$6-0.01y + 0.02y=7.15$
$0.01y=7.15 - 6$
$0.01y=1.15$
$y = 115$
Substitute $y = 115$ into $x + y=600$, we get $x=600 - 115 = 485$.
So, the amount spent on food is $\$485$ and the amount spent on gas is $\$115$.

11.
a. Let $d$ be the amount of data used in GB.
For Carrier A:
If $d\leq3$, the cost $C_A = 35$. If $d>3$, the cost $C_A=35 + 8(d - 3)=35+8d-24=11 + 8d$.
For Carrier B:
If $d\leq5$, the cost $C_B = 55$. If $d>5$, the cost $C_B=55 + 4(d - 5)=55+4d-20=35 + 4d$.
We want to find when $C_A = C_B$.
Case 1: If $d\leq3$, $C_A = 35$ and $C_B\geq55$, so they are not equal.
Case 2: If $3 < d\leq5$, $C_A=11 + 8d$ and $C_B = 55$.
Set $11 + 8d=55$, then $8d=55 - 11=44$, $d = 5.5$ (rejected since $d\leq5$ in this sub - case).
Case 3: If $d>5$, set $11 + 8d=35 + 4d$.
$8d-4d=35 - 11$
$4d=24$
$d = 6$. So, after 6 GB of data use, the two plans cost the same.

b. If $d = 6$:
For Carrier A: $C_A=11+8\times6=11 + 48=59$.
For Carrier B: $C_B=35 + 4\times6=35 + 24=59$.
Since the costs are the same when $d = 6$, either plan can be chosen.

c. Let's consider the cost functions for large values of $d$.
The cost function for Carrier A is $C_A=11 + 8d$ and for Carrier B is $C_B=35 + 4d$.
The difference $C_A - C_B=(11 + 8d)-(35 + 4d)=4d-24$.
When $d>6$, $C_A - C_B>0$, which means $C_A>C_B$.
When $d = 6$, $C_A = C_B$.
When $d<6$, $C_A - C_B<0$, which means $C_AIn the long - run (assuming high data usage), Carrier B is cheaper.