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I'll go through some of the problems on this test to provide solutions. Let me know if there's a specific one you'd like me to start with. Here's a quick overview of how to approach a few selected problems:

### 1. Problem 1: Ratio of Buoys to Gulls
The ratio of buoys to gulls is given as \(14 : 3\). Let \(g\) represent the number of gulls, so the number of buoys is \(\frac{14}{3}g\). We’re told that four times the number of gulls is 6 less than the number of buoys. This can be expressed as:
\[
4g = \frac{14}{3}g - 6
\]
Solving this equation will give the number of gulls.

### 2. Problem 5: Contrapositive of a Statement
The statement is: “If an object is round, then it is a ball.” 
The contrapositive of this statement is: “If an object is not a ball, then it is not round.” This is done by reversing and negating both parts of the original statement.

### 3. Problem 9: Simplify \(( \sqrt{7} - 3 )( \sqrt{7} + 3 )\)
This problem involves simplifying a product of binomials in the form \((a - b)(a + b) = a^2 - b^2\).
For this problem:
\[
( \sqrt{7} - 3 )( \sqrt{7} + 3 ) = (\sqrt{7})^2 - 3^2 = 7 - 9 = -2
\]

### 4. Problem 13: Find the Length of Diagonal \( AC \) in the Rectangular Solid
For a rectangular solid with sides of lengths \( a \), \( b \), and \( c \), the diagonal \( d \) across the solid can be calculated using the formula:
\[
d = \sqrt{a^2 + b^2 + c^2}
\]
Here, the dimensions are 2, 3, and 6. Plugging these in gives:
\[
d = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7
\]

### 5. Problem 20: Surface Area of a Right Circular Cone
To find the surface area of a cone with slant height \( s \) and radius \( r \), the formula is:
\[
A = \pi r (r + s)
\]
Given \( s = 9 \) cm and \( r = 2 \) cm:
\[
A = \pi \times 2 \times (2 + 9) = \pi \times 2 \times 11 = 22\pi \approx 69.08 \, \text{cm}^2
\]

Would you like further detail on any of these problems, or help with others on the page?

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### Relative Questions to Expand Understanding:
1. How do you determine the contrapositive of a conditional statement in logic?
2. What is the formula for the surface area of a cone, and how is it derived?
3. How can you use proportions to solve for unknown quantities in word problems involving ratios?
4. Why does the difference of squares formula apply to the simplification in problem 9?
5. How do you find the diagonal of a rectangular solid using the Pythagorean theorem?

### Tip
When solving problems with ratios or proportions, try cross-multiplying to clear any fractions for a simpler equation structure.

A set of math problems including ratios, simplifying algebraic expressions, geometry, and logic questions. For example: 'The ratio of buoys to gulls was 14 to 3. Four times the number of gulls was 6 less than the number of buoys. How many gulls were there?'

This set of math problems covers a variety of topics including ratios, algebraic equations, simplifying expressions, geometric shapes, and logical reasoning. Learn how to solve problems involving the Pythagorean theorem, surface areas of cones, and algebraic simplifications.

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