Questions: The spinner of the compass is two congruent isosceles triangles connected by their bases as shown in the diagram. The base of each of these triangles is 2 centimeters and the legs are 5 centimeters. If the metal used to construct the spinner costs 13.25 per square centimeter, how much will it cost to make this part of the compass? Round to the nearest cent. Cost =

Solution Steps

To find the height \( h \) of the isosceles triangle, we use the Pythagorean theorem. The height divides the triangle into two right triangles, where the legs are \( \frac{base}{2} = 1 \) cm and \( h \), and the hypotenuse is the leg of the triangle, which is 5 cm. Thus, we have: \[ h = \sqrt{leg^2 - \left(\frac{base}{2}\right)^2} = \sqrt{5^2 - 1^2} = \sqrt{25 - 1} = \sqrt{24} = 2\sqrt{6} \]

The area \( A \) of one isosceles triangle can be calculated using the formula: \[ A = \frac{1}{2} \times base \times height = \frac{1}{2} \times 2 \times h = h \] Substituting the height we found: \[ A = 2\sqrt{6} \]

Since there are two congruent triangles, the total area \( A_{total} \) is: \[ A_{total} = 2 \times A = 2 \times 2\sqrt{6} = 4\sqrt{6} \] The cost \( C \) to construct the spinner is given by: \[ C = A_{total} \times cost\_per\_sq\_cm = 4\sqrt{6} \times 13.25 \] Calculating this gives: \[ C = 53\sqrt{6} \] Finally, rounding the total cost to the nearest cent results in: \[ C \approx 129.82 \]

Final Answer