Problem of water bottles

Question

Suppose that the predicted high temperatures for the next 7 days are given in the table below:

A) Express the high temperature, H, as a function of the day, t.

 A local shop has determined that the number of bottles of water they sell is a function of the day’s high temperature, given by N=3H−80, where H is the day’s high temperature. 

B) How many water bottles will be sold on day 5? 

C) Use your earlier function and the information given to create a new function that represents the number of bottles of water sold, N, as a function of the day, t.

Make sure you get the same answer for t=5 as you did for part B.

Answer 1

Suppose that the predicted high temperatures for the next 7 days are given in the table above:

A) Express the high temperature, H, as a function of the day, t.

Every day, the high temp (H) increases by 2 (slope: m = 2), thus H(t) is linear: H(t) = mt + b ==> H(t) = 2t + b

To find the y-intercept = b, take any point such as: (t, H(t)) = (1, 87) ==> H(t) = 2t + b ==> 87 = 2(1) + b ==> b = 85

Therefore: H(t) = mt + b ==> H(t) = 2t + 85

C) A local shop has determined that the number of bottles of water they sell is a function of the day’s high temperature, given by N=3H−80, where H is the day’s high temperature. Use your earlier function and the information given to create a new function that represents the number of bottles of water sold, N, as a function of the day, t.

Number of bottles of water they sell is a function of the day’s high temperature, given by: N(H) = 3H − 80

N(H) = 3H − 80 ==> N(H(t)) = 3H(t) − 80 ==> N(H(t)) = 3(2t + 85) − 80 ==> N(H(t)) = 6t + 255 − 80 ==> N(H(t)) = 6t + 175

B) How many water bottles will be sold on day 5?

t = 5 ==> N(H(t)) = 6t + 175 ==> N(H(5)) = 6(5) + 175 ==> N(H(5)) = 30 + 175 ==> N(H(5)) = 205 water bottles

Answer 2

assumption: temperature in degrees Fahrenheit

By examining the rate of change between different points, it can be determined that the rate of change is in fact constant. Therefore, H(t) is a linear function of the form y = mx + b

determine the function that models the data points

find slope

m = (y₂ - y₁)/(x₂ - x₁)

= (97 - 89)/(6 - 2)

= 2 ^oF/day

find H-intercept, choose a point from the table and substitute to solve for b

H = 2t + b

97 = 2(6) + b

b = 85

put it all together

H(t) = 2t + 85

The number of bottles (N) is a function of the day's high temperature (H): N(H) = 3H - 80 <-- given

B) determine the number of bottles sold on the fifth day using the provided function

H(t=5) = 2(5) + 85 = 95; therefore, N(H=95) = 3(95) - 80 = 205 bottles

C) make a composite function of the above two functions that represents the number of bottles of water sold as a function of the day, N(H(t)).

N( H(t) ) = 3( 2t + 85 ) - 80

N(t) = 6t + 175

CHECK:

N(t=5) = 6(5) + 175 = 205 bottles