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Maura is creating a quilt using right triangles. A large right triangle with a side length of 40√12

centimeters has smaller right triangles with side lengths of 5√3
centimeters along it. How many smaller lengths are running along the longer length?

5 answers

To find out how many smaller right triangles with side lengths of \(5\sqrt{3}\) centimeters can fit along the longer side of the larger right triangle with a side length of \(40\sqrt{12}\) centimeters, we will follow these steps:

  1. Simplify \(40\sqrt{12}\): \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \] Therefore, \[ 40\sqrt{12} = 40 \times 2\sqrt{3} = 80\sqrt{3} \]

  2. Determine the number of smaller triangles along the longer length: To find out how many smaller triangles of side length \(5\sqrt{3}\) fit into the longer side \(80\sqrt{3}\), we divide the length of the longer side by the length of the smaller triangle: \[ \text{Number of smaller triangles} = \frac{80\sqrt{3}}{5\sqrt{3}} \] Simplifying this expression: \[ = \frac{80}{5} = 16 \]

Thus, the number of smaller lengths running along the longer length of the larger triangle is \(\boxed{16}\).

are you using division or subtraction?

In the context of your question regarding how many smaller right triangles with side lengths of \(5\sqrt{3}\) centimeters can fit along the longer side of a larger right triangle with a side length of \(40\sqrt{12}\) centimeters, I used division to determine the number of smaller triangles that can fit.

Explanation:

  1. Division is used because we are trying to find out how many times the length of the smaller triangle fits into the length of the larger triangle. This means we take the total length of the larger triangle and divide it by the length of the smaller triangle: \[ \text{Number of smaller triangles} = \frac{\text{Length of larger triangle}}{\text{Length of smaller triangle}} \]

  2. Subtraction, on the other hand, would be used if we were trying to find the difference in lengths between the two triangles, which is not what we are calculating here.

So to clarify, in this problem, we are indeed using division to find out how many smaller triangles fit into the larger triangle. The calculation yielded \( \boxed{16} \) smaller triangles fit along the longer length.

well then, if (40\sqrt{12}\) divided by(5\sqrt{3}\ equals sixteen, how come my math calculator got 48?

Let's break down the calculation step-by-step to clarify where the discrepancy might be coming from. We had established the following:

  1. Simplifying \(40\sqrt{12}\): \[ 40\sqrt{12} = 40 \times \sqrt{4 \times 3} = 40 \times 2\sqrt{3} = 80\sqrt{3} \]

  2. Calculating the number of smaller triangles: To find how many \(5\sqrt{3}\) cm triangles fit along \(80\sqrt{3}\) cm, we perform: \[ \text{Number of smaller triangles} = \frac{80\sqrt{3}}{5\sqrt{3}} \]

  3. Performing the division: \[ \frac{80\sqrt{3}}{5\sqrt{3}} = \frac{80}{5} = 16 \]

Given this breakdown, the calculation indeed shows that \(80\sqrt{3} \div 5\sqrt{3} = 16\).

If your calculator indicated \(48\), it could be possible that the input or the calculation was done incorrectly. Please check the values entered into the calculator. The step-by-step breakdown above should clarify that the correct answer is indeed \(16\).

It is also worth verifying that both the completed calculations are based on the right expressions. If you consider different triangle configurations or inputs into the calculator, it might lead to variations, but for the given lengths, the correct mathematical conclusion, as worked out, is \(16\).