https://file.aiquickdraw.com/mathbot/file/17261834396068bkn61qw.jpg

Let's go through the different types of questions in your image and solve them step by step.

### Estimate to the nearest integer:
1. \(\sqrt{38} \approx 6.16\) → **6**
2. \(\sqrt{53} \approx 7.28\) → **7**
3. \(\sqrt{99} \approx 9.95\) → **10**
4. \(\sqrt{227} \approx 15.07\) → **15**
5. \(\sqrt{26} \approx 5.10\) → **5**
6. \(\sqrt{21} \approx 4.58\) → **5**
7. \(\sqrt{80} \approx 8.94\) → **9**
8. \(\sqrt{510} \approx 22.58\) → **23**
9. \(\sqrt{86.4} \approx 9.29\) → **9**
10. \(\sqrt{45.2} \approx 6.72\) → **7**
11. \(\sqrt{\frac{72}{5}} \approx \sqrt{14.4} \approx 3.79\) → **4**
12. \(\sqrt{\frac{27}{8}} \approx \sqrt{3.375} \approx 1.84\) → **2**

### Order from least to greatest:
13. \(8, 10, \sqrt{61}, \sqrt{71}\)
14. \(\sqrt{45}, 9, 6, \sqrt{63}\)
15. \(\sqrt{50}, 7, \sqrt{44}, 5\)

### Algebra: Estimate the solution to each equation to the nearest integer:
16. \(d^2 = 61\) → \(d \approx \sqrt{61} \approx 7.81\) → **8**
17. \(z^2 = 85\) → \(z \approx \sqrt{85} \approx 9.22\) → **9**
18. \(r^2 = 3.7\) → \(r \approx \sqrt{3.7} \approx 1.92\) → **2**

### Geometry:
19. **Find the radius of a cylinder.**
   The volume \(V = 900 \, \text{cm}^3\), height \(h = 10 \, \text{cm}\), and formula \(r = \sqrt{\frac{V}{30}}\):
   \[
   r \approx \sqrt{\frac{900}{30}} = \sqrt{30} \approx 5.48 \, \text{cm}
   \]
   Estimated radius: **5 cm**.

20. **Estimate the speed of the car using the formula \(s = \sqrt{18d}\)** where \(d = 12 \, \text{feet}\):
   \[
   s = \sqrt{18 \times 12} = \sqrt{216} \approx 14.7 \, \text{miles per hour}
   \]
   Estimated speed: **15 mph**.

### Geometry: Estimate the length of the side for each square:
21. **For area = 40 square inches**: 
   The formula is \(A = s^2\), so \(s = \sqrt{40} \approx 6.32\). Estimated side length: **6 inches**.

22. **For area = 97 square feet**: 
   \(s = \sqrt{97} \approx 9.85\). Estimated side length: **10 feet**.

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Do you want further clarifications or more details on any of these solutions?

Here are 5 related questions:
1. How do you estimate square roots without a calculator?
2. How can you simplify the square root of a fraction?
3. What is the formula for the volume of a cylinder?
4. How is the braking distance of a car related to its speed?
5. How do you order irrational numbers like square roots?

**Tip**: To estimate square roots, find the two closest perfect squares and use them as a guide. For example, \(\sqrt{38}\) is between \(\sqrt{36} = 6\) and \(\sqrt{49} = 7\).

Estimate square roots, order them, solve algebraic and geometry problems based on an image containing various mathematical questions, such as estimating square roots, ordering them, solving algebraic equations, and geometry problems involving the volume of cylinders and area of squares.

This page covers math problems that involve estimating square roots, solving algebraic equations, and geometry questions like finding the radius of a cylinder and the side of a square. Suitable for students in Grades 7-9.

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