Question

The average monthly temperature of the water in a mountain stream is modeled by the function 18sin(π6x−π2)+53, where T(x) is the temperature of the water in °F in month x (x=1 → January). Solve algebraically to find which two months are most likely to give a temperature reading of 62°F.

a) January and July

b) February and August

c) March and September

d) April and October

Answer 1

To find the two months that are most likely to give a temperature reading of 62°F, we need to solve the equation 18sin(π/6x−π/2)+53 = 62. The two months that are most likely to give this temperature reading are March and November.

Step-by-step explanation:

To find the two months that are most likely to give a temperature reading of 62°F, we need to solve the equation 18sin(π/6x−π/2)+53 = 62. Here's how:

1. Subtract 53 from both sides: 18sin(π/6x−π/2) = 9.

2. Divide both sides by 18 to isolate the sine function: sin(π/6x−π/2) = 9/18.

3. Simplify the right side: sin(π/6x−π/2) = 1/2.

4. Find the angle whose sine is 1/2: π/6x−π/2 = π/6 or 5π/6.

5. Solve for x: π/6x = π/6 + π/2 or 5π/6 + π/2. This gives two possible values for x: x = (1/6 + 1/2)/π or (5/6 + 1/2)/π.

6. Simplify and convert to months: x = 3/π or 11/π. These correspond to the months March and November.

Therefore, the two months that are most likely to give a temperature reading of 62°F are March and November.