Question

a jewelry designer plans to make a triangular pendant out of gold wire. the wire cost $31.65 per cm. what is the possible range of cost for the wire? because this is money, round to the hundredths place.
10.5 cm
? cm
2.5 cm
answer attempt 1 out of 2
between $ and $

Explanation:

Step1: Find the range of the third side

For a triangle, the length of the third side \( x \) must satisfy the triangle inequality: the sum of any two sides must be greater than the third side. So we have two inequalities:

1. \( x + 2.5>10.5 \) which gives \( x > 10.5 - 2.5=8 \)
2. \( x<10.5 + 2.5 = 13 \)

Also, \( x+10.5>2.5 \) (always true since \( x>0 \)) and \( 2.5 + 10.5>x \) (which is the same as \( x < 13 \)) and \( 2.5+x>10.5\) (which is \( x > 8 \)) and \( 10.5 + x>2.5\) (always true). So the third side \( x \) is in the range \( 8 < x<13 \)

Step2: Find the range of the perimeter

The perimeter \( P \) of the triangle is \( P=10.5 + 2.5+x=13 + x \)
When \( x>8 \), \( P=13 + x>13 + 8 = 21 \)
When \( x<13 \), \( P=13 + x<13+13 = 26 \)
So the perimeter \( P \) is in the range \( 21 < P<26 \)

Step3: Find the range of the cost

The cost \( C \) is the perimeter times the cost per cm (\( \$31.65 \) per cm). So \( C = 31.65\times P \)
When \( P > 21 \), \( C>31.65\times21 \)
\( 31.65\times21=31.65\times(20 + 1)=31.65\times20+31.65\times1 = 633+31.65=664.65 \)
When \( P < 26 \), \( C<31.65\times26 \)
\( 31.65\times26=(30 + 1.65)\times26=30\times26+1.65\times26 = 780+42.9 = 822.9 \)

Answer:

Between $\boldsymbol{664.65}$ and $\boldsymbol{822.90}$