Parsiparts in the Dance-a-Thon recelve donations per hour they dance or one-time donations. The table shovs the toial donations $y$ (in doflars) a studert earns dancing $x$ number of hours. The votat amount a professor earns dancing $x$ mimber of mours is representa by the equation $y=20 x+500$
\begin{tabular}{cc}
Number of hours, $x$ & Total donations, $)$ (n dollars) \\
3 & 435 \\
6 & 570 \\
9 & 705 \\
12 & 840
\end{tabular}

Select all the statements that are true.
Both participants receive $\$ 660$ when they dance for 8 hours.
The student receives $\$ 145$ for every hour of dancing.
The professor received $\$ 500$ in one-time donations.
The student receives more donations when they both dance for 2 hours.
The professor receives more donations when they both dance for 10 hours.

Step 1 ：Calculate the rate at which the student receives donations per hour using the given table.

Step 2 ：Use the equation \(y=20x+500\) to calculate the total donations the professor receives for the given number of hours.

Step 3 ：Compare the student's donations to the professor's donations for 8 hours, 2 hours, and 10 hours.

Step 4 ：Determine the truth of the statements based on the calculations.

Step 5 ：\( \text{Rate per hour for the student} = \frac{\text{Donations at 6 hours} - \text{Donations at 3 hours}}{\text{6 hours} - \text{3 hours}} = \frac{570 - 435}{6 - 3} = \frac{135}{3} = 45 \)

Step 6 ：\( \text{Student donations for 8 hours} = 45 \times 8 = 360 \)

Step 7 ：\( \text{Professor donations for 8 hours} = 20 \times 8 + 500 = 660 \)

Step 8 ：\( \text{Student donations for 2 hours} = 45 \times 2 = 90 \)

Step 9 ：\( \text{Professor donations for 2 hours} = 20 \times 2 + 500 = 540 \)

Step 10 ：\( \text{Student donations for 10 hours} = 45 \times 10 = 450 \)

Step 11 ：\( \text{Professor donations for 10 hours} = 20 \times 10 + 500 = 700 \)

Step 12 ：\( \text{Statement 1 is false because both participants do not receive } \$660 \text{ when they dance for 8 hours.} \)

Step 13 ：\( \text{Statement 2 is false because the student receives } \$45 \text{ per hour, not } \$145. \)

Step 14 ：\( \text{Statement 3 is true because the professor received } \$500 \text{ in one-time donations.} \)

Step 15 ：\( \text{Statement 4 is false because the professor receives more donations when they both dance for 2 hours.} \)

Step 16 ：\( \text{Statement 5 is true because the professor receives more donations when they both dance for 10 hours.} \)

Step 17 ：\( \boxed{\text{The professor received } \$ 500 \text{ in one-time donations.}} \)

Step 18 ：\( \boxed{\text{The professor receives more donations when they both dance for 10 hours.}} \)