QuestionThando's bag is 35cm x 35cm x 30cm. How many 27cm x 27cm x 3.5cm pizza boxes fit in it? After a 15% size increase, can they still fit?

Studdy Solution

STEP 1

Assumptions1. The dimensions of the heat-insulated bag are length35cm, width35cm, and height30cm. . The dimensions of the pizza box are length27cm, width27cm, and height3.5cm.
3. The pizza boxes fit perfectly inside the bag without any wasted space.
4. The pizza boxes do not deform when stacked.
5. The dimensions of the pizza box will increase by15% in all directions.

STEP 2

First, we need to find the volume of the heat-insulated bag and the pizza box. The volume of a rectangular prism (which both the bag and the box are) is found by multiplying the length, width, and height.
$Volume = Length \\times Width \\times Height$

STEP 3

Now, plug in the given values for the dimensions of the heat-insulated bag to calculate its volume.
$Volume_{bag} =35\,cm \\times35\,cm \\times30\,cm$

STEP 4

Calculate the volume of the heat-insulated bag.
$Volume_{bag} =35\,cm \\times35\,cm \\times30\,cm =36,750\,cm^3$

STEP 5

Now, plug in the given values for the dimensions of the pizza box to calculate its volume.
$Volume_{box} =27\,cm \\times27\,cm \\times3.5\,cm$

STEP 6

Calculate the volume of the pizza box.
$Volume_{box} =27\,cm \\times27\,cm \\times3.5\,cm =2,565.5\,cm^3$

STEP 7

Now that we have the volumes of both the bag and the box, we can divide the volume of the bag by the volume of the box to find out how many boxes can fit into the bag.
$Number\,of\,boxes = Volume_{bag} / Volume_{box}$

STEP 8

Plug in the values for the volumes of the bag and the box to calculate the number of boxes.
$Number\,of\,boxes =36,750\,cm^3 /2,565.5\,cm^3$

STEP 9

Calculate the number of boxes that can fit into the bag.
$Number\,of\,boxes =36,750\,cm^3 /2,565.5\,cm^3 \approx14$ So, Hotizzas could fit approximately14 pizza boxes into the heat-insulated bag.

STEP 10

Now, we need to calculate the new dimensions of the pizza box after a15% increase in all directions. We can do this by multiplying the original dimensions by.15 (which represents a15% increase).
$New\,dimension = Original\,dimension \\times.15$

STEP 11

Calculate the new dimensions of the pizza box.
$New\,length =27\,cm \\times.15 =31.05\,cm$ $New\,width =27\,cm \\times.15 =31.05\,cm$ $New\,height =3.5\,cm \\times.15 =4.025\,cm$

STEP 12

Now, we need to check if the new pizza box can still fit into the heat-insulated bag. We can do this by comparing the new dimensions of the pizza box with the dimensions of the bag. If any of the new dimensions of the pizza box are larger than the corresponding dimension of the bag, then the new box will not fit.

STEP 13

Compare the new dimensions of the pizza box with the dimensions of the bag.
The new length and width of the pizza box (31.05 cm) are smaller than the length and width of the bag (35 cm), and the new height of the pizza box (.025 cm) is smaller than the height of the bag (30 cm). Therefore, the new pizza box will still fit into the heat-insulated bag.

STEP 14

Now, we need to find out how many of the new pizza boxes can fit into the bag. First, calculate the volume of the new pizza box by multiplying the new length, width, and height.
$Volume_{new\,box} = New\,length \\times New\,width \\times New\,height$

STEP 15

Plug in the values for the new dimensions of the pizza box to calculate its volume.
$Volume_{new\,box} =31.05\,cm \\times31.05\,cm \\times4.025\,cm$

STEP 16

Calculate the volume of the new pizza box.
$Volume_{new\,box} =31.05\,cm \\times31.05\,cm \\times4.025\,cm =3,873.6\,cm^3$

STEP 17

Now, divide the volume of the bag by the volume of the new box to find out how many new boxes can fit into the bag.
$Number\,of\,new\,boxes = Volume_{bag} / Volume_{new\,box}$

STEP 18

Plug in the values for the volumes of the bag and the new box to calculate the number of new boxes.
$Number\,of\,new\,boxes =36,750\,cm^3 /3,873.6\,cm^3$

STEP 19

Calculate the number of new boxes that can fit into the bag.
$Number\,of\,new\,boxes =36,750\,cm^3 /3,873.6\,cm^3 \approx9$ So, Hotizzas could fit approximately9 of their new, larger pizza boxes into the heat-insulated bag.

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