Answer :
Sure, let's work through the question step-by-step:
(a) Completing the Table:
We are given the relation [tex]\( F = \frac{9}{5}C + 32 \)[/tex] to convert temperatures from degrees Celsius (°C) to degrees Fahrenheit (°F). Let's fill in the table for the given Celsius values:
1. For [tex]\( C = 0 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 0 + 32 = 32 \][/tex]
2. For [tex]\( C = 5 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 5 + 32 = 41 \][/tex]
3. For [tex]\( C = 10 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 10 + 32 = 50 \][/tex]
4. For [tex]\( C = 15 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 15 + 32 = 59 \][/tex]
5. For [tex]\( C = 20 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 20 + 32 = 68 \][/tex]
6. For [tex]\( C = 25 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 25 + 32 = 77 \][/tex]
7. For [tex]\( C = 30 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 30 + 32 = 86 \][/tex]
Now, the completed table looks like this:
[tex]\[
\begin{array}{|l|c|c|c|c|c|c|c|}
\hline
{ }^{\circ} C & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline
{ }^{\circ} F & 32 & 41 & 50 & 59 & 68 & 77 & 86 \\
\hline
\end{array}
\][/tex]
(b) Drawing the Graph:
To draw the graph, you would plot these Celsius and Fahrenheit pairs on a graph paper. Use a scale where 2 cm represents 10 Fahrenheit units on the vertical axis and 5 Celsius units on the horizontal axis. Plot the points: (0, 32), (5, 41), (10, 50), (15, 59), (20, 68), (25, 77), and (30, 86) and draw a straight line connecting them.
(c) Finding Celsius when [tex]\( F = 55 \)[/tex]:
To find the temperature in Celsius when [tex]\( F = 55 \)[/tex], we rearrange the formula to solve for [tex]\( C \)[/tex]:
[tex]\[ 55 = \frac{9}{5}C + 32 \][/tex]
Subtract 32 from both sides:
[tex]\[ 23 = \frac{9}{5}C \][/tex]
Multiply both sides by [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[ C = 23 \times \frac{5}{9} \approx 12.78 \][/tex]
So, when [tex]\( F = 55 \)[/tex], the temperature in Celsius is approximately [tex]\( 12.78^\circ C \)[/tex].
(d) Interpretation of the Slope:
The slope of the relation [tex]\( F = \frac{9}{5}C + 32 \)[/tex] is [tex]\(\frac{9}{5}\)[/tex] or 1.8. This means that for every 1 degree increase in Celsius, the temperature in Fahrenheit increases by 1.8 degrees. This is the rate of change between Fahrenheit and Celsius temperatures.
(a) Completing the Table:
We are given the relation [tex]\( F = \frac{9}{5}C + 32 \)[/tex] to convert temperatures from degrees Celsius (°C) to degrees Fahrenheit (°F). Let's fill in the table for the given Celsius values:
1. For [tex]\( C = 0 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 0 + 32 = 32 \][/tex]
2. For [tex]\( C = 5 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 5 + 32 = 41 \][/tex]
3. For [tex]\( C = 10 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 10 + 32 = 50 \][/tex]
4. For [tex]\( C = 15 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 15 + 32 = 59 \][/tex]
5. For [tex]\( C = 20 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 20 + 32 = 68 \][/tex]
6. For [tex]\( C = 25 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 25 + 32 = 77 \][/tex]
7. For [tex]\( C = 30 \)[/tex]:
[tex]\[ F = \frac{9}{5} \times 30 + 32 = 86 \][/tex]
Now, the completed table looks like this:
[tex]\[
\begin{array}{|l|c|c|c|c|c|c|c|}
\hline
{ }^{\circ} C & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline
{ }^{\circ} F & 32 & 41 & 50 & 59 & 68 & 77 & 86 \\
\hline
\end{array}
\][/tex]
(b) Drawing the Graph:
To draw the graph, you would plot these Celsius and Fahrenheit pairs on a graph paper. Use a scale where 2 cm represents 10 Fahrenheit units on the vertical axis and 5 Celsius units on the horizontal axis. Plot the points: (0, 32), (5, 41), (10, 50), (15, 59), (20, 68), (25, 77), and (30, 86) and draw a straight line connecting them.
(c) Finding Celsius when [tex]\( F = 55 \)[/tex]:
To find the temperature in Celsius when [tex]\( F = 55 \)[/tex], we rearrange the formula to solve for [tex]\( C \)[/tex]:
[tex]\[ 55 = \frac{9}{5}C + 32 \][/tex]
Subtract 32 from both sides:
[tex]\[ 23 = \frac{9}{5}C \][/tex]
Multiply both sides by [tex]\(\frac{5}{9}\)[/tex]:
[tex]\[ C = 23 \times \frac{5}{9} \approx 12.78 \][/tex]
So, when [tex]\( F = 55 \)[/tex], the temperature in Celsius is approximately [tex]\( 12.78^\circ C \)[/tex].
(d) Interpretation of the Slope:
The slope of the relation [tex]\( F = \frac{9}{5}C + 32 \)[/tex] is [tex]\(\frac{9}{5}\)[/tex] or 1.8. This means that for every 1 degree increase in Celsius, the temperature in Fahrenheit increases by 1.8 degrees. This is the rate of change between Fahrenheit and Celsius temperatures.
