The national weather service is recording the temperatures on a cold day. The temperature at 6 A.M. is -14^circ F The temperature increases an average of (3)/(4)F each hour until noon , and then it drops an average of -1(1)/(4)^circ F each hour until 5 P M. Which of the following statements are true? Select all that apply. The temperature at 5 P.M. is -15(3)/(4)^circ F The change in the temperature from noon to 5 P.M. is -6(1)/(4)^circ F At noon, the temperature is -9(1)/(2)^circ F The total change in the temperature from 6 A.M. to 5 P.M. is -2(1)/(4)^circ F The change in the temperature from 6 A M. to noon is (1)/(8)F.

To determine which statements are true, we need to calculate the temperature at different times throughout the day.<br /><br />Given information:<br />- The temperature at 6 A.M. is $-14^{\circ }F$.<br />- The temperature increases an average of $\frac {3}{4}F$ each hour until noon.<br />- The temperature drops an average of $-1\frac {1}{4}^{\circ }F$ each hour until 5 P.M.<br /><br />Step 1: Calculate the temperature at noon.<br />The temperature increases by $\frac {3}{4}F$ each hour from 6 A.M. to noon (6 hours).<br />Temperature increase from 6 A.M. to noon = $\frac {3}{4}F \times 6 = \frac {18}{4}F = 4\frac {1}{2}F$<br /><br />Temperature at noon = Temperature at 6 A.M. + Temperature increase from 6 A.M. to noon<br />Temperature at noon = $-14^{\circ }F + 4\frac {1}{2}F = -9\frac {1}{2}^{\circ }F$<br /><br />Step 2: Calculate the temperature at 5 P.M.<br />The temperature drops by $-1\frac {1}{4}^{\circ }F$ each hour from noon to 5 P.M. (5 hours).<br />Temperature decrease from noon to 5 P.M. = $-1\frac {1}{4}^{\circ }F \times 5 = -\frac {5}{4}F = -1\frac {1}{4}F$<br /><br />Temperature at 5 P.M. = Temperature at noon + Temperature decrease from noon to 5 P.M.<br />Temperature at 5 P.M. = $-9\frac {1}{2}^{\circ }F + (-1\frac {1}{4}F) = -10\frac {3}{4}^{\circ }F$<br /><br />Step 3: Calculate the change in temperature from noon to 5 P.M.<br />Change in temperature from noon to 5 P.M. = Temperature at 5 P.M. - Temperature at noon<br />Change in temperature from noon to 5 P.M. = $-10\frac {3}{4}^{\circ }F - (-9\frac {1}{2}^{\circ }F) = -1\frac {1}{4}F$<br /><br />Step 4: Calculate the total change in temperature from 6 A.M. to 5 P.M.<br />Total change in temperature from 6 A.M. to 5 P.M. = Temperature at 5 P.M. - Temperature at 6 A.M.<br />Total change in temperature from 6 A.M. to 5 P.M. = $-10\frac {3}{4}^{\circ }F - (-14^{\circ }F) = 3\frac {1}{4}F$<br /><br />Step 5: Calculate the change in temperature from 6 A.M. to noon.<br />Change in temperature from 6 A.M. to noon = Temperature at noon - Temperature at 6 A.M.<br />Change in temperature from 6 A.M. to noon = $-9\frac {1}{2}^{\circ }F - (-14^{\circ }F) = 4\frac {1}{2}F$<br /><br />Therefore, the true statements are:<br />- The temperature at 5 P.M. is $-10\frac {3}{4}^{\circ }F$<br />- The change in the temperature from noon to 5 P.M. is $-1\frac {1}{4}^{\circ }F$<br />- At noon, the temperature is $-9\frac {1}{2}^{\circ }F$<br />- The total change in the temperature from 6 A.M. to 5 P.M. is $3\frac {1}{4}F$<br />- The change in the temperature from 6 A.M. to noon is $4\frac {1}{2}F$