Make a model of a rectangular prism that is not a cube and has a surface area of 62 cm squared. What are the measurements of this rectangular prism?
I know the formula, is 2ab+2bc+2ac=sa, and I know the answer to this problem... a=5, b=3, and c=2... I did this by trial and error, and I know that there has to be a simpler way. I am trying to teach my 5th grader how to find the solution to this problem, and I don't know how to do it without just doing trial and error. If someone could explain this to me so that I could teach him, I'd really appreciate it. Thanks in advance.
Hmmm. To make things easier, assume you have integer dimensions.
Also, noting that the sum of one of each of the parallel faces is half the area, or 31 in this case.
So, you want three numbers that have factors and add up to 31.
There's still some trial and error, since there are three variables, and only one equation.
You know that at least two of the numbers have to be odd, since you want to end up with three products that add up to 31, and odd number.
So, starting small,
if a=2, then you could have b=3, giving you 2*3 + 2c+3c = 31, or c=5. First Try!
You won't always be so lucky, but by analyzing the information a bit, you can usually cut down on the false starts.
To find a rectangular prism's measurements when given its surface area, you can simplify the process by using some algebraic techniques.
Let's start by understanding the formula for the surface area of a rectangular prism:
Surface Area (SA) = 2ab + 2bc + 2ac
Where a, b, and c are the dimensions of the rectangular prism.
In this case, the surface area is given as 62 cm². Let's substitute this into the formula:
62 = 2ab + 2bc + 2ac
Now, we want to find the measurements (values of a, b, and c) that satisfy this equation.
To simplify the equation, let's divide both sides by 2:
31 = ab + bc + ac
Now, we can try to factorize the equation using a common term:
31 = a(b + c) + bc
Next, we want to find factors of 31, the prime number. The factors of 31 are 1 and 31. To find values for b and c, we need to consider all possible combinations of b and c that give us the factors of 31.
Let's try the combinations:
1 = a(31) + 1 (b + c must equal 1)
Here, a = 1, b = 0, and c = 1. However, a and c should be nonzero values, so this combination is not valid.
31 = a(1) + 31 (b + c must equal 31)
Here, a = 31, b = 0, and c = 31. But again, b should be nonzero, so this combination is also invalid.
Since we couldn't find the valid combinations using the factors of 31, we need to explore the prime factorization of 31. The prime factorization of 31 is 31 = 31 * 1 or 1 * 31.
Let's consider the combination where b + c = 31 and a = 1:
31 = 1(b + c) + bc
31 = 1(b + c) + (b)(c)
Now, we need to find the values of b and c that satisfy this equation. We can create a table to help us:
| b | c | b + c | bc |
------------------------
1 | 1 | 30 | 31 | 30 |
2 | 2 | 29 | 31 | 58 |
...|...|...|.......|... |
29 |29 | 2 | 31 | 58 |
30 |30 | 1 | 31 | 30 |
------------------------
As you can see from the table, the combination that works is when b = 2, c = 29, and a = 1.
Therefore, the measurements of the rectangular prism with a surface area of 62 cm² are: a = 1 cm, b = 2 cm, and c = 29 cm.
By systematically analyzing the equation and the factors, we found the correct measurements without trial and error. You can guide your 5th grader through this process to help them understand how to find a solution more efficiently.