1. Whitewater Recreation Area proposed the following rate increases for kayak rentals.

Proposed Kayak Rental Rate Increase

Rate Type Current Rate Increased Rate

Initial Charge $32 $35

Hourly Rate $5 $7

(a) Write a variable expression showing the total cost to rent a kayak at the current rate. Write a variable expression showing the total cost to rent a kayak at the increased rate. Define the variable.

(b) Use the variable expressions to calculate the cost to rent a kayak for 8 hours at the current rate and at the increased rate. Show all your work.

(c) Calculate the percent increase in the total cost to rent a kayak for 8 hours if the proposed rates are approved. Show all your work.

(d) Emmie paid $62 for a kayak rental at the current rate. Write and solve an equation to determine the number of hours she rented the kayak. Show all your work.

Step-by-step explanation:

Let's start with the basic concept. to rent a kyak you pay an initial charge then every hour you pay more. If we look at the initial conditions this is saying to rent a kyak you pay 32 dollars then pay 5 more every hour. So hour 0 is 32, then hour 1 gets you to 32+5 = 37, then hour 2 gets you to 32+2*5=32+10=42 and so on.

This allows us to write the answer to the first part of (a). If we say P is price and h is number of hours, you can keep track of the price with the equation P = 32 + 5h. Check it with the few examples I gave with 0, 1 and 2 hours. The second part of (a) is figured out very similarly and gets you P = 35 + 7h.

(b) So just like you could find the values for 0, 1 and two hours by putting those values in for h, you can d the same here and just put 8 in So Current is 32 + 5 (8) = 72 and The increased rate would get 91. Make sure you show your work though, plug in and go through all the arithmetic.

(c) A percent increase is asking how much something increases by in percent form. For instance if you have 100 dollars and you increase it by 50%, or half, you take half of the original amount and add it. so 100/2 = 50, so increasing 100 by 50% is 100+50 = 150. Another way to write perents is decimal form. for 50% the decimal form is. 5, or for something like 5% it's. 05. You just move the decimal place to the left twice, or in other words divide it by 100.

Anyway, to find the total of increasing something by a percent you can do this formula. P + Pd = P (1+d) where you could use either one, and P is the price you got from the last formula and d is the decimal form of the percent you are increasing by. To try it out increase 100 by 50%, so you'd plug in 100 + 100*.5 = 100 (1+.5) = 100 (1.5) = 150. Now that this was explained we can get the percent increase. Or in other words, we know the end result and we know P, we don't know d though.

P (1+d) = 91

72 (1+d) = 91

(1+d) = 91/72

d = 91/72-1

d =.26

Now, remember I said d is the decimal form of the percent, which you get by moving the decimal to the left twice. Well to go the other way you move the decimal the other way. so the percent is 26%

(d) This one's super easy. It's taking the formula for the current rate P = 32 + 5h where we know P is 62 and we want to find h

62 = 32 + 5h

30 = 5h

6 = h

Let me know if anything didn't make sense.