10. Given the function $$ y=2 \cos \left(x+30^{\circ}\right) $$ for $$ 0^{\circ} \leq x \leq 540^{\circ} $$, state: i. The period ii. The phase angle 11. Determine the interquartile range for the following set of numbers $$ 10,15,14,17,12,13 $$

Question

UpStudy AI · Accepted Answer

Let's address each question step by step.

---

### **Question 10**

**Given the function:** $$ y = 2 \cos \left(x + 30^{\circ}\right) $$ for $$ 0^{\circ} \leq x \leq 540^{\circ} $$

**i. The Period**

The general form of a cosine function is:
$$ y = A \cos(Bx + C) + D $$

- **Amplitude:** $$ |A| $$
- **Period:** $$ \frac{360^{\circ}}{B} $$
- **Phase Shift:** $$ -\frac{C}{B} $$

For the given function:
- $$ A = 2 $$
- $$ B = 1 $$
- $$ C = 30^{\circ} $$

**Calculating the Period:**
$$ \text{Period} = \frac{360^{\circ}}{B} = \frac{360^{\circ}}{1} = 360^{\circ} $$

**Answer:** The period of the function is **360 degrees**.

---

**ii. The Phase Angle**

**Calculating the Phase Shift:**
$$ \text{Phase Shift} = -\frac{C}{B} = -\frac{30^{\circ}}{1} = -30^{\circ} $$

The negative sign indicates a shift to the **left**.

**Answer:** The phase angle (phase shift) is **-30 degrees**, meaning the graph is shifted 30 degrees to the left.

---

### **Question 11**

**Determine the Interquartile Range (IQR) for the dataset:** $$ 10, 15, 14, 17, 12, 13 $$

**Steps to Calculate the IQR:**

1. **Arrange the Data in Ascending Order:**
   $$ 10, 12, 13, 14, 15, 17 $$

2. **Find the Median (Q2):**
   - Since there are 6 data points (even number), the median is the average of the 3rd and 4th values.
   $$ \text{Median (Q2)} = \frac{13 + 14}{2} = 13.5 $$

3. **Find the First Quartile (Q1):**
   - Lower half: $$ 10, 12, 13 $$
   - Median of the lower half: $$ 12 $$

4. **Find the Third Quartile (Q3):**
   - Upper half: $$ 14, 15, 17 $$
   - Median of the upper half: $$ 15 $$

5. **Calculate the IQR:**
   $$ \text{IQR} = Q3 - Q1 = 15 - 12 = 3 $$

**Answer:** The interquartile range of the dataset is **3**.

---
**Question 10:** - **i. Period:** 360 degrees - **ii. Phase Angle:** -30 degrees **Question 11:** - **Interquartile Range (IQR):** 3

Given the function for , state:
i. The period
ii. The phase angle

Determine the interquartile range for the following set of numbers

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Trigonometry Questions

Given the function for , state: i. The period ii. The phase angle Determine the interquartile range for the following set of numbers