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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.Scarlett works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh kilogram each, and are shipped in a container that weighs kilograms. Large ones, on the other hand, weigh kilograms apiece, and are shipped in a container that weighs kilograms. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. How many helicopters would fit in either container? What would the total weight be?If either container holds _____ helicopters, it will weigh a total of _____ kilograms once it is packed for shipping.
Full solution
Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.Scarlett works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh kilogram each, and are shipped in a container that weighs kilograms. Large ones, on the other hand, weigh kilograms apiece, and are shipped in a container that weighs kilograms. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. How many helicopters would fit in either container? What would the total weight be?If either container holds _____ helicopters, it will weigh a total of _____ kilograms once it is packed for shipping.
- Define Variables: Let's define the variables:Let be the number of small helicopters in a container.Let be the number of large helicopters in a container.The total weight of a container with small helicopters is (since each small helicopter weighs kilogram and the container itself weighs kilograms).The total weight of a container with large helicopters is (since each large helicopter weighs kilograms and the container itself weighs kilograms).We are given that the total weight of both types of containers is the same. Therefore, we can write the following equation:
- Solve for x: Now, let's solve for one of the variables. We can solve for in terms of or in terms of . Let's solve for :
Now we have an equation for in terms of . - Set Up Equations: Since we want to find the number of helicopters that would fit in either container, we need to set up a second equation that represents the condition that the containers will have the same shipping weight. We already have the equation for the total weight of each container, so we can use the equation we just found to express in terms of and substitute it into the first equation to find the value of .
- Substitute Expression: Substitute the expression for into the first equation:This simplifies to , which is always true, indicating that we have an infinite number of solutions. This means that for any number of large helicopters , there is a corresponding number of small helicopters that will make the containers weigh the same. However, we need to find the specific number of helicopters that would fit in either container, so we need to re-evaluate our approach.
- Correct Approach: We realize that we made a mistake in our approach. We should have set up two separate equations, one for the total weight of the small helicopter container and one for the total weight of the large helicopter container, and then set them equal to each other. Let's correct this:For small helicopters: Total weight = For large helicopters: Total weight = Since the total weights are equal, we set the two equations equal to each other:
