A theatre +-decides to change the shape of its popcorn container from a regular box to a right regular pyramid, as shown in the following figure, and charge only half as much. If the containers are the same height and the tops are the same size, is this a bargain for the customer? Explain.
calculate the volume for each shape
and compare based on the new cost
they show the shapes but don't give any numbers to work with
To determine whether it's a bargain for the customer, we need to compare the volumes of the two containers: the regular box and the right regular pyramid.
Let's consider the regular box first. The volume of a box is given by the formula V = l × w × h, where l is the length, w is the width, and h is the height. Since the regular box has a rectangular base, its volume is V_box = l × w × h_box.
Now let's move on to the right regular pyramid. The volume of a right regular pyramid is given by the formula V = (1/3) × B × h, where B is the area of the base and h is the height. Since the tops of both containers are the same size, the base of the pyramid is also the same size as the base of the box. Therefore, the volume of the right regular pyramid is V_pyramid = (1/3) × B_box × h_pyramid.
The price change mentioned in the question is that the pyramid container is now half the price of the box container. Therefore, we can compare the volumes of the two containers and see if the cost difference matches the volume difference.
To do this, let's take the ratio of the volumes of the pyramid container to the box container:
V_ratio = V_pyramid / V_box
= ((1/3) × B_box × h_pyramid) / (l × w × h_box)
Since the height of both containers is the same, h_pyramid = h_box, we can simplify the ratio:
V_ratio = ((1/3) × B_box × h_box) / (l × w × h_box)
= (1/3) × B_box / (l × w)
Now, we know that the new pyramid container is half the price of the box container. So the price ratio will be 1/2, which means:
Price_ratio = Price_pyramid / Price_box
= (1/2)
= 1/2
Since the price ratio should be the same as the volume ratio, we have:
V_ratio = Price_ratio
Substituting the respective equations, we get:
(1/3) × B_box / (l × w) = 1/2
To determine whether it's a bargain for the customer, we can rearrange this equation to solve for B_box:
B_box = (3/2) × l × w
This tells us that the area of the base of the box container should be equal to (3/2) times the length multiplied by the width of the rectangular base.
Now, if we compare this with a regular rectangular box, we find that the base area for the pyramid container is greater than the base area for the box container. This means that the pyramid container actually contains more popcorn than the regular box container, even though it is being sold at half the price.
Therefore, it can be concluded that changing the shape of the popcorn container to a right regular pyramid and charging only half as much is indeed a bargain for the customer, as they are getting more popcorn for the same price.