Part 1The temperature in a certain city is modeled byUpper T left parenthesis x right parenthesis equals 37 plus 21 sine left bracket StartFraction 2 pi Over 365 EndFraction left parenthesis x minus 91 right parenthesis right bracketT(x)=37+21sin2π365(x−91)where T(x) is the temperature in degrees Fahrenheit on day x, with x = 1 corresponding to January 1 and xequals=365 corresponding to December 31. (a) Use a calculator to estimate the temperature on March 15 left parenthesis day 74 right parenthesisMarch 15 (day 74).(b) Use a calculator to estimate the temperature on March 1616. Part 1(a) The temperature on March 15 left parenthesis day 74 right parenthesisMarch 15 (day 74) will be about [input]enter your response here degrees°. (Do not round until the final answer. Then round to the nearest integer as needed.)

Question

Part 1The temperature in a certain city is modeled byUpper T left parenthesis x right parenthesis equals 37 plus 21 sine left bracket StartFraction 2 pi Over 365 EndFraction left parenthesis x minus 91 right parenthesis right bracketT(x)=37+21sin2π365(x−91)where​ T(x) is the temperature in degrees Fahrenheit on day​ x, with x​ = 1 corresponding to January 1 and xequals=365 corresponding to December 31. ​(a) Use a calculator to estimate the temperature on March 15 left parenthesis day 74 right parenthesisMarch 15 (day 74).​(b) Use a calculator to estimate the temperature on March 1616. Part 1​(a) The temperature on March 15 left parenthesis day 74 right parenthesisMarch 15 (day 74) will be about [input]enter your response here degrees°. ​(Do not round until the final answer. Then round to the nearest integer as​ needed.)

BlackTom AI · Accepted Answer

We are given the temperature model T(x) = 37 + 21 sin( (2π/365) (x − 91) ), where x is the day of the year. The task asks for the temperature on March 15, which corresponds to day x = 74.

First, determine the argument of the sine function for x = 74:
(2π/365)(x − 91) = (2π/365)(74 − 91) = (2π/365)(−17) = −34π/365.

Next, approximate the angle in radians: −34π/365 ≈ −0.2929 radians. The sine of this small negative angle is approximately sin(−0.2929) ≈ −0.289.

Now multiply by the amplitude 21 to see how much the sine term shifts the baseline: 21 × (−0.289) ≈ −6.07.

Add this to the baseline temperature 37: 37 + (−6.07) ≈ 30.93.

Rounding instructions say not to round until the final step, and then round to the nearest integer as needed. The intermediate value 30.93 rounds to 31 when rounded to the nearest whole number.

A few quick checks: since the sine term is negative at day 74, the temperature is below the midline 37, which aligns with getting a value near 31. If one used a calculator, the sine of −34π/365 would yield a result close to −0.289, leading to the same final estimate through the same arithmetic.

Thus, the estimated temperature on March 15 is approximately 31 degrees Fahrenheit when rounded to the nearest integer.

Homework:Section 6.3 Homework

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