Question

each of the following temperature ranges is in degrees fahrenheit. use the formula $f = \frac{9}{5}c + 32$ to find the corresponding temperature ranges in degrees celsius. answer using interval notation and assume endpoints are included. round values to the nearest integer, if necessary.\
a. $5^{\circ}$ to $14^{\circ}$\
b. $41^{\circ}$ to $77^{\circ}$\
c. $-13^{\circ}$ to $-4^{\circ}$\
d. $113^{\circ}$ to $131^{\circ}$

Explanation:

Part a: \( 5^\circ \) to \( 14^\circ \) Fahrenheit

Step 1: Solve the formula for \( C \)

We start with the formula \( F = \frac{9}{5}C + 32 \). To solve for \( C \), we first subtract 32 from both sides: \( F - 32 = \frac{9}{5}C \). Then we multiply both sides by \( \frac{5}{9} \): \( C = \frac{5}{9}(F - 32) \).

Step 2: Find the lower bound (F = 5)

Substitute \( F = 5 \) into the formula for \( C \):
\( C = \frac{5}{9}(5 - 32) = \frac{5}{9}(-27) = -15 \)

Step 3: Find the upper bound (F = 14)

Substitute \( F = 14 \) into the formula for \( C \):
\( C = \frac{5}{9}(14 - 32) = \frac{5}{9}(-18) = -10 \)

Step 1: Use the formula \( C = \frac{5}{9}(F - 32) \) (already derived)

Step 2: Find the lower bound (F = 41)

Substitute \( F = 41 \) into the formula:
\( C = \frac{5}{9}(41 - 32) = \frac{5}{9}(9) = 5 \)

Step 3: Find the upper bound (F = 77)

Substitute \( F = 77 \) into the formula:
\( C = \frac{5}{9}(77 - 32) = \frac{5}{9}(45) = 25 \)

Step 1: Use the formula \( C = \frac{5}{9}(F - 32) \)

Step 2: Find the lower bound (F = -13)

Substitute \( F = -13 \) into the formula:
\( C = \frac{5}{9}(-13 - 32) = \frac{5}{9}(-45) = -25 \)

Step 3: Find the upper bound (F = -4)

Substitute \( F = -4 \) into the formula:
\( C = \frac{5}{9}(-4 - 32) = \frac{5}{9}(-36) = -20 \)

Answer:

\([-15, -10]\)

Part b: \( 41^\circ \) to \( 77^\circ \) Fahrenheit