Question

a jewelry designer plans to make a triangular pendant out of gold wire. the wire costs $31.65 per centimeter. what is the possible range of costs for the wire?
the wire can cost more than $\square$ and less than $\square$.

Explanation:

Step1: Find the range of the third - side of the triangle

According to the triangle - inequality theorem: \(a + b>c\) and \(|a - b|\(10.5−2.5 < c<10.5 + 2.5\), so \(8 < c<13\).

Step2: Find the range of the perimeter of the triangle

The perimeter \(P=a + b + c\), where \(a = 2.5\) and \(b = 10.5\). So \(P=13 + c\).
Since \(8 < c<13\), then \(13+8 < P<13 + 13\), \(21 < P<26\).

Step3: Calculate the cost range

The cost \(C\) of the wire is \(C = 31.65P\) (because the cost is \(\$31.65\) per centimeter).
When \(P = 21\), \(C=31.65\times21=664.65\).
When \(P = 26\), \(C=31.65\times26 = 822.9\).

Answer:

The wire can cost more than \(\$664.65\) and less than \(\$822.9\).