Geometry Word Problem

Question

Sara builds floor lamps that fit into boxes that are 4 feet by 2 feet by 2 feet. The bed of her pickup truck has a maximum clearance that is 54 inches wide by ten feet long and a hard cover that lies flat at 30 inches from the bottom. How many lamps can Sara fit into her truck at one time?

 

A. 4

B. 6

C. 7

D. 8

Answer 1

Hello Chandler,

 

This is a simple Volume problem,

 

what we wish to find is the number of lamps that can fit into the bed of a truck

 

lets first find the volume each lamp takes up

 

recall that the volume of a rectangle is

 

V = Length x Width x Height = LWH = (4 ft) x (2 ft ) x (2ft) = 16 ft³

 

now lets find the total volume in the bed of the truck

 

V = LWH = (10 ft) x (54 in) x (30 in)

 

we have a unit problem with both the width and height so lets convert them

 

recall that,

12 in = 1 ft or 1 in = 1/12 ft

 

so,

54 in = (54 in) x (1 ft) / (12 in) = 4.5 ft         and  30 in = (30 in) x (1 ft) / (12 in) = 2.5 ft

now that the units are fixed we can calculate the correct volume

 

V= LWH= (10 ft) x (4.5 ft) x (2.5 ft) = 112.5 ft³

 

now that we have the total volume of the truck bed and the volume of each lamp box lets find how many of those lamp boxes can fit in to the bed of the truck

 

so the total number of boxes

 

Boxes = Total Volume of truck bed / Volume of each box

 

B = (112.5 ft³) / (16 ft³) = 7.03 boxes

 

so the maximum number of boxes that will fit into the bed of the truck is C. 7

 

 

Answer 2

I think there's an error in your problem's wording. First, let's convert those inch measurements to feet:

 

54" = 4.5'

30" = 2.5'

 

That means the truck bed is 10 × 4.5 × 2.5 feet. Since the height is only a little over 2 feet, we'll never be able to stack any lamps on top of each other and only the 2-foot side of the lamps will ever be able to fit there, so we can ignore that dimension and just focus on fitting 4 × 2 lamps into a 10 × 4.5 truck bed.

 

 

There are a few ways you could do this, and they're difficult to explain in words:

 

1. Slide two lamps into the truck one behind the other so their 4-foot sides lie along the truck's 10-foot side. Now you've filled a 2 × 8 foot space. Do the same with one more pair of lamps so you've filled a 4 × 8 foot space. Finally, put one lamp oriented the other way at the bottoms of those lamps so its 2-foot side finishes filling the truck's 10-foot side. Now you've filled a 4 × 10 foot space, which is pretty much all the space you have. You can't fit any more lamps. That's a total of 5 lamps.

 

2. Slide as many lamps as you can into the truck so that their 4-foot sides lie along the truck's 4.5-foot side. The truck is 10 feet long, so you'll be able to fit 5 2-foot-wide lamps in that way. Now you've again filled a 4 × 10 foot space and can't fit any more lamps. That's a total of 5 lamps.

 

3. Slide three lamps into the truck so that their 4-foot sides lie along the truck's 4.5-foot side, like in (2). Now you've filled a 6 × 4 foot space. Now, slide two more lamps into the truck oriented the other way, with their 2-foot sides along the truck's 4.5-foot side. You've again filled a 4 × 10 foot space and can't fit any more lamps. That's a total of

5 lamps.

 

 

So, it seems to me like the answer must be 5 lamps. I can't find anything wrong with my logic here. I think you just haven't been give the right answer as a possible answer choice.