Question

A jewelry designer is creating a necklace that has a length of
32.97
32.97 inches. To determine the placement of different jewels the designer uses a circle to represent the necklace. The designer will place sapphires from point
K
K to point
L
L and diamonds from point
L
L to point
M
.
M. The design and some of the measurements that represent the necklace are shown.



m
∠
K
H
L
=
42
°
m∠KHL=42° and
m
∠
L
H
M
=
23
°
m∠LHM=23°







What is the difference between the length, in inches, of the necklace portion that has sapphires and the length in inches, of the necklace portion that has diamonds? Use
3.14
3.14 for
π
.
π. Round your answer to the nearest hundredth.

Answer 1

To find the lengths of the portions of the necklace that have sapphires (from point K to point L) and diamonds (from point L to point M), we first need to find the angle in degrees that each portion represents relative to the total 360 degrees of the circle (representing the entire necklace).

1. **Calculate the lengths of the necklace portions:**
The necklace is modeled as a circle with a total length of 32.97 inches. The formulas below will help compute the lengths:

- Length of arc (portion) = (Central angle / Total angle) * Circumference

Where the circumference of the circle (C) is given as the total length of the necklace:

\[
C = 32.97 \text{ inches}
\]

The total angle (in a circle) = 360 degrees.

2. **Calculate the length of the sapphires portion (K to L)**:
The central angle \( m∠KHL = 42° \).

Using the formula:

\[
\text{Length of sapphires} = \left(\frac{42}{360}\right) \times 32.97
\]

Calculate:

\[
\text{Length of sapphires} = \left(\frac{42}{360}\right) \times 32.97 \approx 4.85 \text{ inches}
\]

3. **Calculate the length of the diamonds portion (L to M)**:
The central angle \( m∠LHM = 23° \).

Using the formula:

\[
\text{Length of diamonds} = \left(\frac{23}{360}\right) \times 32.97
\]

Calculate:

\[
\text{Length of diamonds} = \left(\frac{23}{360}\right) \times 32.97 \approx 2.02 \text{ inches}
\]

4. **Calculate the difference between the lengths**:
Now we find the difference in length:

\[
\text{Difference} = \text{Length of sapphires} - \text{Length of diamonds} \approx 4.85 - 2.02
\]

Calculate:

\[
\text{Difference} \approx 2.83 \text{ inches}
\]

5. **Final Answer**:
The difference between the length of the necklace portion that has sapphires and the length of the portion that has diamonds is approximately

\[
\boxed{2.83} \text{ inches}
\] when rounded to the nearest hundredth.