Question

1. apply math models 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?

10.5 cm ? cm
2.5 cm

Explanation:

Step1: Find the range of the third side length

According to the triangle - inequality theorem, for a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a = 10.5\) cm and \(b=2.5\) cm. Then \(|10.5 - 2.5|=8\) cm and \(10.5 + 2.5 = 13\) cm. So the length of the third side \(x\) satisfies \(8\lt x\lt13\).

Step2: Find the range of the perimeter

The perimeter \(P\) of the triangle is \(P=a + b + x\), where \(a = 10.5\) cm and \(b = 2.5\) cm. Substitute \(a\) and \(b\) into the perimeter formula: \(P=10.5 + 2.5+x=13 + x\).
Since \(8\lt x\lt13\), then \(13 + 8\lt P\lt13+13\), so \(21\lt P\lt26\).

Step3: Find the range of the cost

The cost \(C\) of the wire is \(C = 31.65P\) (where \(P\) is the perimeter in cm).
When \(P = 21\), \(C_1=31.65\times21=(30 + 1.65)\times21=30\times21+1.65\times21=630+34.65 = 664.65\).
When \(P = 26\), \(C_2=31.65\times26=(30 + 1.65)\times26=30\times26+1.65\times26=780 + 42.9=822.9\).

Answer:

The possible range of costs for the wire is \(\$664.65\lt C\lt\$822.9\)