Question

Tina is raising money for a charity. She makes either 18 telephone callsper day or 10 home visits per day. Tina wants to contact at least 100 peoplenext week. This situation can be represented by the following system ofinequalities, where x= number of days making telephone calls and y=number of days making home visits. Which option is a true statement?*

Answer 1

The variables we have are:

x --> represents the number of days making phone calls

y--> represents the number of days of home visits.

The first given inequation is:

`x+y\leq7`

This represents that the total number of days is less than or equal to 7, and that is because she only has 1 week (7 days).

The second given inequality is:

`18x+10y\ge100`

This inequality is related to the number of people that she contacts. Since she contacts 18 people by making phone calls per day, 18x is the total number of people that she contacted on the telephone in x number of days.

And 10y is the total number of people that she contacted by making home visits in y number of days.

Now that we know what the two inequalities represent, we can move on to reviewing the options to have and find the correct answer.

Analyzing the given options:

Option A. Tina can reach her goal by making home visits only.

To find if this is true we suppose that Tina makes home visits for 7 days, which means:

`\begin{gathered} y=7 \\ x=0 \end{gathered}`

Since she only makes home visits y=7 and there are no days in which she makes phone calls so x=0.

We substitute these values into the second equation to see if she can reach her goal of at least 100 people:

`18(0)+10(7)\ge100`

Solving the operations:

`\begin{gathered} 0+70\ge10 \\ 70\ge100 \end{gathered}`

Is clear that 70 is not equal to or greater than 100, thus she cannot reach her goal by making only home visits. Thus, option A is not correct.

Option B. Tina can reach her goal with only 2 days of telephone calls.

If she only works 2 days, and those two days are telephone calls:

`\begin{gathered} x=2 \\ y=0 \end{gathered}`

Again we substitute in the second inequality:

`\begin{gathered} 18(2)+10(0)\ge100 \\ 36+0\ge100 \\ 36\ge100 \end{gathered}`

She will only contact 36 people and she needs 100 or more, thus, option B is not correct.

Option C. If Tina does home visits for 4 days, she only needs 2 days of telephone calls.

In this case, let's see if with 4 days of home visits y=4, and 2 days of telephone calls x=2, she reaches her goal.

Substituting the values of x and y into the second inequality:

`18(2)+10(4)\ge100`

solving the operations:

`\begin{gathered} 36+40\ge100 \\ 76\ge100 \end{gathered}`

76 people are not equal to or greater than 100 people and thus, she will not reach her goal with option C.

Option D. If Tina makes telephone calls for 5 days, she only needs 1 day of home visits.

5 days of telephone calls --> x=5

1 day of home visits --> y=1

Substituting x and y into the second inequality:

`\begin{gathered} 18(5)+10(1)\ge100 \\ 90+10\ge100 \\ 100\ge100 \end{gathered}`

We can see that in this case, she will contact exactly 100 people, which was her goal.

Answer. D