A commercial bakery uses a mixing bowl that initially contains some flour. A machine adds … — SAT

A commercial bakery uses a mixing bowl that initially contains some flour. A machine adds flour to the bowl at a constant rate. After the machine has been running for $8$ minutes, the total mass of the flour in the bowl is $14$ kilograms. After the machine has been running for $20$ minutes, the total mass of the flour in the bowl is $29$ kilograms. The relationship between the total mass of the flour in the bowl, $M$ , in kilograms, and the time the machine has been running, $t$ , in minutes, is linear. What is the mass of the flour, in kilograms, in the bowl before the machine starts running?

Answer: 4 kg

Answer

The relationship between the total mass of the flour,

M

, and the time,

t

, is linear and can be represented by the equation

M = mt + b

, where

m

is the rate at which flour is added and

b

is the initial mass of the flour in the bowl. Using the two given points,

(8, 14)

and

(20, 29)

, the slope

m

is calculated as

m = \frac{29 - 14}{20 - 8} = \frac{15}{12} = 1.25

kilograms per minute. Substituting

m = 1.25

and the point

(8, 14)

into the linear equation gives

14 = 1.25(8) + b

, which simplifies to

14 = 10 + b

. Solving for

b

yields

b = 4

. Therefore, the mass of the flour in the bowl before the machine starts running is

4

kilograms.

Step-by-Step Solution

Find the rate of change (slope) of the linear relationship.

The rate is

1.25

kilograms per minute.

The slope of a linear relationship represents the constant rate at which flour is added to the bowl.

Set up the linear equation and solve for the y-intercept.

The initial mass is

4

kilograms.

The y-intercept represents the initial mass of the flour in the bowl at

t = 0

minutes.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question

Answer

Step-by-Step Solution

Key Concept

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