Math  /  Algebra

QuestionBotanical Gardens The membership fee for joining a gardening association is $24\$ 24 per year. A local botanical garden charges members of the gardening association $3\$ 3 for admission to the garden. Nonmembers of the association are charged $6\$ 6. After how many visits to the garden is the total cost for members, including the membership fee, the same as the total cost for nonmembers?

Studdy Solution

STEP 1

What is this asking? How many times does a person have to visit the botanical garden so that paying for a yearly membership is the same as just paying the higher non-member price per visit? Watch out! Don't forget to include the membership fee when calculating the total cost for members!

STEP 2

1. Define the cost functions
2. Set up an equation
3. Solve for the number of visits

STEP 3

Let's **define** Cm(x)C_m(x) as the **total cost** for *members* after xx visits.
Since members pay a $24\$24 yearly fee plus $3\$3 per visit, the function is Cm(x)=24+3xC_m(x) = 24 + 3 \cdot x.
The **initial value** is the $24\$24 membership fee, and the **rate of change** is $3\$3 per visit.

STEP 4

Now, let's **define** Cn(x)C_n(x) as the **total cost** for *non-members* after xx visits.
Non-members pay $6\$6 per visit, so the function is Cn(x)=6xC_n(x) = 6 \cdot x.
Here, the **rate of change** is $6\$6 per visit.

STEP 5

We want to find the number of visits, xx, where the total cost for members equals the total cost for non-members.
So, we set the two cost functions equal to each other: Cm(x)=Cn(x)C_m(x) = C_n(x).
This gives us the equation 24+3x=6x24 + 3 \cdot x = 6 \cdot x.

STEP 6

To **isolate** xx, we first want to get all the terms with xx on one side of the equation.
Let's *subtract* 3x3 \cdot x from *both* sides of the equation: 24+3x3x=6x3x24 + 3 \cdot x - 3 \cdot x = 6 \cdot x - 3 \cdot x.
This simplifies to 24=3x24 = 3 \cdot x.

STEP 7

Now, we want to get xx by itself.
We can do this by *dividing* both sides of the equation by 33: 243=3x3\frac{24}{3} = \frac{3 \cdot x}{3}.

STEP 8

This gives us the **final result**: x=8x = 8.

STEP 9

After **8 visits**, the total cost for members and non-members will be the same.
Specifically, that cost will be $48\$48.

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