Question

a paper hat is folded into the shape of a kite, as shown. (3x + 5) in. (5x + 1) in. (2x + 1) in. 5 in. what total length of ribbon is needed to line the two long sides of the hat? 10 in. 11 in. 20 in. 22 in.

Explanation:

Step1: Recall properties of a kite

In a kite, one pair of adjacent sides are equal, and the other pair of adjacent sides are equal. From the diagram, we can see that the sides with lengths \((2x + 1)\) in and \(5\) in are one pair of adjacent sides (since they are adjacent to the right angles), and the sides with lengths \((3x + 5)\) in and \((5x + 1)\) in are the other pair of adjacent sides. So we can set \((2x + 1)=5\) to find the value of \(x\).

Step2: Solve for \(x\)

Given \(2x + 1 = 5\), subtract \(1\) from both sides: \(2x=5 - 1=4\). Then divide both sides by \(2\): \(x=\frac{4}{2} = 2\).

Step3: Find the length of the long sides

Now we find the length of the sides \((3x + 5)\) and \((5x + 1)\) by substituting \(x = 2\).

For \((3x+5)\): Substitute \(x = 2\), we get \(3\times2+5=6 + 5=11\) in.

For \((5x + 1)\): Substitute \(x = 2\), we get \(5\times2+1=10 + 1=11\) in.

Step4: Find the total length of the two long sides

The two long sides each have length \(11\) in, so the total length is \(11+11 = 22\) in.

Answer:

22 in.