Question

1. A farmer is monitoring the water content of the soil during a dry summer.
The water content of the soil is decreasing at a rate of 2.5% per week. If
a certain soil sample contains 200 grams of water on July 1st, which
equation describes the relationship between y, amount of water in grams,
and t, time in weeks after July 1st?
A y = 200(0.75)'
B. y = 200(0.975)
C. y = 200 -0.0251
D= 200 -0.251

Answer 1

B) (since 1-0.025=0.975)

Answer 2

The equation that describes the relationship between y, amount of water in grams,  and t, time in weeks after July 1st is $y= f(t) = 200(97.5)^t$

Exponential Regression equation

An exponential regression equation shows the decrement of something from the original value after every period of time such that the decrement is at a constant rate.

$y = x(1-r)^n$

where,

y is the value after t period of time at a decrement rate of r.

Given to us

• The water content of the soil is decreasing at a rate of 2.5% per week.

• certain soil sample contains 200 grams of water on July 1st,

As we know that for such problems we use exponential regression equation.

Water content in the begining

As at the begining of the 1st July the soil had water content of 200 grams therefore, the value of the constant in the exponential equation will be 200.

Rate of decrement in water Content

the value of water content in the soil is decreasing by 2.5% after every week, therefore, the water left in the soil will be 97.5% of the previous week, therefore, the value of rate of r in the equation will be 2.5%. while the value of n in the equation will be t.

Substituting the values in the formula of exponential regression equation,

$f(t) = 200(1-0.025)^t$

where, t is the number of weeks after 1st July.

Note: On 1st july the value of t will be 0.

further the equation can be solved as,

$f(t) = 200(97.5)^t$

$y = f(t) = 200(97.5)^t$

Hence, the equation that describes the relationship between y, amount of water in grams,  and t, time in weeks after July 1st is $y= f(t) = 200(97.5)^t$.

Learn more about Exponential Regression equation:

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