Question 1093879
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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