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Hello could you please help me with question number four?

Hello could you please help me with question number four?-example-1
Hello could you please help me with question number four?-example-1
Hello could you please help me with question number four?-example-2

1 Answer

6 votes

Answer

- Question 1:

The equation for the number of memberships is:


M=550(1+0.03)^t

- Question 2:

The number of memberships in 2020 will be 964

Step-by-step explanation

Problem Statement

We are told that a gym sold 550 memberships in 2001 and that the number of memberships sold has increased by 3% every year since then.

We are asked to find:

1. The equation for the number of memberships sold for t years after 2001.

2. The number of memberships sold by 2020.

Solution

Question 1:

- The question tells us that the number of memberships sold in 2001 is 550. This is the initial value of our equation.

- We are told that the number of memberships increases 3% every year since 2001.

- This implies that:


\begin{gathered} In\text{ year 2001, the number of memberships sold }=550 \\ \\ In\text{ year 2002, 1 year from 2001, the number of memberships sold}=Amount\text{ sold in 2001 + 3\% of the Amount sold in 2001} \\ T\text{he number of memberships sold in 2002}=550+(3\text{ \% }*550)=550+550(0.03)=550(1+0.03) \\ \\ In\text{ year 2003, 2 years from 2001, the number memberships sold}=\text{Amount sold in 2002 + 3\% of the Amount sold in 2002} \\ The\text{ number of memberships sold in 2003}=(550+(3\text{ \% }*550))+(3\text{ \%}*(550+(3\text{ \% }*550)) \\ =550\mleft(1+0.03\mright)+0.03\mleft(550\mleft(1+0.03\mright)\mright) \\ \text{ We can factorize this expression as follows:} \\ =550\mleft(1+0.03\mright)(1+0.03) \\ =550(1+0.03)^2 \\ \\ In\text{ year 2004},\text{ 3 years from 2001, the number of memberships sold}=Amount\text{ sold in 2003 + 3\% of the Amount sold in 2003} \\ =550(1+0.03)^2+3\text{ \% (}550(1+0.03)^2) \\ =550(1+0.03)^2+0.03(550(1+0.03)^2) \\ \text{ We can factorize again:} \\ 550(1+0.03)^2(1+0.03) \\ =550(1+0.03)^3 \end{gathered}

- Observing the above illustration, we can see a pattern. This pattern is that:


\begin{gathered} \text{The amount of memberships sold }t\text{ years after 2001 is given by:} \\ 550(1+0.03)^t \end{gathered}

- Thus, we have the equation for the number of memberships sold t years after 2001 to be:


\begin{gathered} M=550(1+0.03)^t \\ \text{where,} \\ M=\text{ Number of memberships sold} \\ t=\text{The number of years after 2001} \end{gathered}

Question 2:

- We are asked to calculate the number of memberships the gym would sell by 2020.

- First, we need to find the value of t since it is the only variable in our formula. We have already defined t to be the number of years after 2001.

- Thus, t can be gotten as follows:


\begin{gathered} t=2020-2001 \\ t=19 \end{gathered}

- Now that we have t, we can find the number of memberships as follows:


\begin{gathered} M=550(1+0.03)^(19) \\ M=964.43\approx964\text{ memberships} \\ \\ (\text{ We rounded down the number from 964.43 to 964 because we cannot have 0.43 memberships)} \end{gathered}

- Thus, the number of memberships in 2020 will be 964

Final Answer

- Question 1:

The equation for the number of memberships is:


M=550(1+0.03)^t

- Question 2:

The number of memberships in 2020 will be 964

User Vladimir Romanov
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