The curved surface area, $$ A $$, of a cone of height $$ h $$ and base radius $$ r $$ is $$ \pi r \sqrt{h^{2}+r^{2}} $$. a Make $$ h $$ the subject of the formula. b Find the height of a cone of area $$ 550 \mathrm{~cm}^{2} $$ and base radius 7 cm , taking $$ \pi $$ to be $$ \frac{22}{7} $$.

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UpStudy AI · Accepted Answer

Sure, let's tackle both parts of the problem step by step.

### **Given:**
- Curved Surface Area of a cone, $$ A = \pi r \sqrt{h^{2} + r^{2}} $$
- $$ \pi = \frac{22}{7} $$ (for part b)
  
---

### **a. Making $$ h $$ the Subject of the Formula**

We need to solve the formula for $$ h $$ in terms of $$ A $$ and $$ r $$.

**Step 1:** Start with the given formula:
$$
A = \pi r \sqrt{h^{2} + r^{2}}
$$

**Step 2:** Isolate the square root term by dividing both sides by $$ \pi r $$:
$$
\frac{A}{\pi r} = \sqrt{h^{2} + r^{2}}
$$

**Step 3:** Square both sides to eliminate the square root:
$$
\left( \frac{A}{\pi r} \right)^2 = h^{2} + r^{2}
$$

**Step 4:** Subtract $$ r^{2} $$ from both sides to solve for $$ h^{2} $$:
$$
h^{2} = \left( \frac{A}{\pi r} \right)^2 - r^{2}
$$

**Step 5:** Take the square root of both sides to solve for $$ h $$:
$$
h = \sqrt{ \left( \frac{A}{\pi r} \right)^2 - r^{2} }
$$
$$
\boxed{h = \sqrt{\,\left(\dfrac{A}{\pi r}\right)^2 - r^2\,}}
$$

---

### **b. Finding the Height $$ h $$ When $$ A = 550 \ \text{cm}^2 $$ and $$ r = 7 \ \text{cm} $$**

**Given:**
- $$ A = 550 \ \text{cm}^2 $$
- $$ r = 7 \ \text{cm} $$
- $$ \pi = \frac{22}{7} $$

**Step 1:** Use the formula derived in part a:
$$
h = \sqrt{ \left( \frac{A}{\pi r} \right)^2 - r^{2} }
$$

**Step 2:** Substitute the known values into the formula:
$$
h = \sqrt{ \left( \frac{550}{\frac{22}{7} \times 7} \right)^2 - 7^{2} }
$$

**Step 3:** Simplify the denominator:
$$
\frac{22}{7} \times 7 = 22
$$
$$
h = \sqrt{ \left( \frac{550}{22} \right)^2 - 49 }
$$

**Step 4:** Divide 550 by 22:
$$
\frac{550}{22} = 25
$$
$$
h = \sqrt{25^2 - 49}
$$

**Step 5:** Compute $$ 25^2 $$ and subtract 49:
$$
25^2 = 625
$$
$$
625 - 49 = 576
$$
$$
h = \sqrt{576}
$$

**Step 6:** Find the square root of 576:
$$
h = 24 \ \text{cm}
$$

$$
\boxed{h = 24 \ \text{cm}}
$$

---

### **Summary:**
- **a.** The height $$ h $$ in terms of $$ A $$ and $$ r $$ is:
  $$
  h = \sqrt{\,\left(\dfrac{A}{\pi r}\right)^2 - r^2\,}
  $$
  
- **b.** For a cone with a curved surface area of $$ 550 \ \text{cm}^2 $$ and a base radius of $$ 7 \ \text{cm} $$, the height is:
  $$
  h = 24 \ \text{cm}
  $$
a. The height $$ h $$ is $$ \sqrt{\left(\frac{A}{\pi r}\right)^2 - r^2} $$. b. The height of the cone is 24 cm.

The curved surface area, \( A \), of a cone of height \( h \) and base radius \( r \) is \( \pi r \sqrt{h^{2}+r^{2}} \). a Make \( h \) the subject of the formula. b Find the height of a cone of area \( 550 \mathrm{~cm}^{2} \) and base radius 7 cm , taking \( \pi \) to be \( \frac{22}{7} \).

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