You are planning the layout for a new supermarket!!

Question

you are planning the layout for a new supermarket.
you are planning the layout for a new supermarket. You need to know how much space should be reserved in the parking lot and in the store entrance for shopping carts. shopping carts can "nest", or fit partially inside each other.

a) if one cart has a lenght of 3 feet, and a stack of 12 nested carts is 16.75 feet long, how much of each cart is inside another when they are "nested"?

b)what woould the length of a 20-cart stack be?

c) how many carts would it take to make a nested stack 34.25 feet long.

d)write an equation that gives the length, L, of a stack of n shopping carts.

e) the supermarket will have some shopping carts inside the store and some outside in the parking lot. if the maximum length of the inside shopping cart space is 50 feet, how many nested shopping carts can fit in the space in three rows? (remember to consider space to pull out carts from the stack)

f) if the supermarket has a total of 200 carts and the maximum number of carts is stored inside at night, how many carts must be kept outside?

Answer 1

First off we need to set some variables.

 

Let's let C = number of carts and N = the number of carts that are inside another cart. N will just be C - 1 since the cart on the end of the row cannot be in another cart since it is on the front of the row.

 

Scroll down to page 8 of this PDF to see a picture of the setup.

http://www.nps.k12.nj.us/SCI/wp-content/uploads/sites/67/2015/06/Rising-8th-Graders-or-Entering-Algebra-I.pdf

 

Now on to part a).

How long would a row of carts be if they were not nested? 3 ft x 12 carts = 36 ft.

How long is the row with nested carts? 16.75 ft (this is a given).

What's the difference in the two measurements? 36 - 16.75 = 19.25 ft. The nested row is 19.25 feet shorter. When we divide this by the 11 nested carts we will find out how much each cart is in another cart. 19.25/11 = 1.75 ft.

 

Each cart is 1.75 feet inside another cart.

 

Now on to part b).

First find the length of an un-nested row: 20 carts x 3 ft = 60 ft.

Now multiply the 1.75 feet we got earlier by the number of carts there are minus 1: 1.75 x (20 - 1) = 33.25. This number is the amount of cart nested in another cart totaled.

Now subtract this number from 60 to get 60 - 33.25 = 26.75 ft. This is the row length of 20 nested carts.

 

Part c): Do this after finishing part d) since d) is easier.

 

Part d): We let C = the number of carts earlier and we let N = C - 1. We let L be the total length of the row.

 

Notice what we did in part a), we subtracted the 1.75 x N from 3 x C to get L.

 

In equation form this is: 3C - 1.75N = L or 3C - 1.75(C - 1) = L. Simplified this is: 1.25C + 1.75 = L.

 

This is the equation: 1.25C + 1.75 = L.

 

Part c): Just solve for C. 1.25C + 1.75 = 34.25 ==> 34.25 - 1.75 = 1.25C ==> C = 26 carts. You need 26 carts.

 

Part e): This part is a little bit of trial and error. Remember, you cannot only part of a cart and you cannot have a row that is longer than 50 feet long. The best thing to do is try guessing a number of carts and plug it into the equation from part d).

 

Let's try C = 30 carts: 1.25 x 30 + 1.75 = 39.25 feet long. That's good, but we can probably put more carts in there. Let's try C = 40 carts: 1.25 x 40 + 1.75 = 51.75 feet long. That's too long. Our answer is somewhere between 30 and 40 carts. Let's try 38 (because I did it for you and it's 2 carts fewer, which should yield a slightly smaller number).

 

C = 38 carts: 1.25 x 38 + 1.75 = 49.25. This is right near the edge. You cannot fit any more carts in a single row than this. Since there are 3 rows multiply the 38 carts in one row by 3 to get a total of 114 carts in 3 rows. You can keep 114 carts inside.

 

Part f): Since there are 200 carts and 114 of them can stay inside, how many are outside?

 

200 - 114 = 86 carts outside at night.

 

There you go.

 

Hope this helps!

 

Edit: For the equation we made in part d), C should be n since that is how the question wanted the answer. This means the answer for d) should be 1.25n + 1.75 = L. The same holds throughout the rest of the problem answer.