1. **Problem 1: Find the correct formula for $F$ given $C = \frac{5}{9}(F-32)$.** The original formula relates Celsius ($C$) and Fahrenheit ($F$) as: $$C = \frac{5}{9}(F - 32)$$ To solve for $F$, multiply both sides by $\frac{9}{5}$: $$\frac{9}{5}C = F - 32$$ Then add 32 to both sides: $$F = \frac{9}{5}C + 32$$ Comparing with the options: - A. $F = \frac{5}{9}(C-32)$ (incorrect) - B. $F = \frac{9}{5}(C-32)$ (incorrect) - C. $F = \frac{9}{5}(C+32)$ (incorrect) - D. $F = \frac{5}{9}(C+32)$ (incorrect) None of the options exactly match the correct formula $F = \frac{9}{5}C + 32$. However, option C is close but has $C+32$ inside the fraction, which is wrong. **Answer: None of the given options are correct.** 2. **Problem 2: Calculate $S$ given $S = 2\pi (r+n)$, $S=440$, $r=7$, and $\pi = \frac{22}{7}$. Find $n$.** Given: $$S = 2\pi (r + n) = 440$$ $$r = 7$$ $$\pi = \frac{22}{7}$$ Substitute values: $$440 = 2 \times \frac{22}{7} \times (7 + n)$$ Simplify: $$440 = \frac{44}{7} (7 + n)$$ Multiply both sides by 7: $$440 \times 7 = 44 (7 + n)$$ $$3080 = 44 (7 + n)$$ Divide both sides by 44: $$\frac{3080}{44} = 7 + n$$ $$70 = 7 + n$$ Subtract 7: $$n = 70 - 7 = 63$$ **Answer: $n = 63$ (Option A).** 3. **Problem 3: Simplify the expression $\frac{4y^2 - 13y - 12}{4y - 3}$.** We want to simplify: $$\frac{4y^2 - 13y - 12}{4y - 3}$$ First, factor the numerator $4y^2 - 13y - 12$. Find two numbers that multiply to $4 \times (-12) = -48$ and add to $-13$. These numbers are $-16$ and $3$. Rewrite numerator: $$4y^2 - 16y + 3y - 12$$ Group terms: $$(4y^2 - 16y) + (3y - 12)$$ Factor each group: $$4y(y - 4) + 3(y - 4)$$ Factor out common binomial: $$(4y + 3)(y - 4)$$ So: $$\frac{(4y + 3)(y - 4)}{4y - 3}$$ Since denominator is $4y - 3$, which is not a factor of numerator, the expression cannot be simplified further. **Answer: The expression does not simplify to any of the options given.** **Summary:** - Problem 1: None of the options match the correct formula $F = \frac{9}{5}C + 32$. - Problem 2: $n = 63$ (Option A). - Problem 3: Expression does not simplify to any given option.