Question 1093879: Estabon poured himself a hot beverage that had a temperature of 203
degreesF then set it on the kitchen table to cool. The temperature of the kitchen was a constant 68degreesF. If the drink cooled to 182degreesF in 3 minutes, how long will it take for the drink to cool to 104 degreesF?
(Do not round until the final answer. Then round to the nearest minute as needed.)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! assuming that the cooling of the drink is linear, then you get:
(203 - 182)/3 = 21/3 = 7 degrees of cooling per minute.
that's the slope of a linear equation of y = mx + b.
m is the slope.
b is the y-intercept, otherwise known as the value of y when x = 0.
so your equation is y = mx + b which becomes y = -7x + b because the slope is going down because the temperature is dropping at a constant rate of 7 degrees per minute.
your coordinate points in this equation are (3,182).
those are the points that you know.
another point is (0,203).
(0,203) tells you that when x = 0, y = 203.
x is the time in minutes.
y is the temperature.
(3,182) tells you that when x = 3, y = 182.
you already know the y-intercept because you know the value of y when x is equal to 0.
it's 203, but we'll solve for it anyway and we should get that value.
we'll use the point (3,182).
the equation is y = -7x + b
replace y with 182 and x with 3 and we get 182 = -7*3 + b.
simplify to get 182 = -21 + b.
add 21 to both sides of the equation to get 203 = b
that's our y-intercept.
the equation is therefore y = -7x + 203.
we want to know how long it will take for the temperature to reach 104.
since y is the temperature, then we get 104 = -7x + 203.
subtract 203 from both sides of the equation to get -99 = -7x.
divide both sides of the equation by -7 to get -99/-7 = x
solve for x to get x = -99/-7 = 14.14285714 minutes.
it takes the liquid 14.14285714 minutes to go from 203 degrees to 104 degrees.
it takes the liquid 3 minutes to go from 203 to 182 degrees.
it therefore takes the liquid 14.14285714 - 3 = 11.14285714 minute to go from 182 to 104 degrees.
round to the nearest minutes to get 3 minutes from 203 to 182 and 11 minutes from 182 to 104 for a total of 14 minutes.
we solved this using linear algebra, but we could have solved just using logic.
203 - 182 = 21 divided by 3 = 7 degrees of loss per minute.
182 - 104 = 78 / 7 = 11.14285714 minutes to go from 182 to 104.
total time from 203 to 104 is 11.14285714 + 3 = 14.14285714 minutes.
we can graph this equation and it will show us where the critical points are.
here's the graph.
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