Question

A large open meadow near your city is shrinking in size because developers have started building new homes there.

Answer 1

Answer:

    $t=-10\log 0.32$

Explanation:

Equation:

         $A=1,562.5\times 10^{(-0.1t)}$

This is a decaying exponential function, because the exponent is negative.

Then, the larger the value of t (number of months), the smaller A (the number of hectares).

The factor 1,562.5 is the initial number of hecares, when t = 0.

You must find the value of t when A equals 500, thus solve the equation:

     $500=1,562.5\times 10^{(-0.1t)$

Divide both sides by 1,562.5

     $\dfrac{500}{1,562.5}=10^{(-0.1t)}$

    $0.32=10^{(-0.1t)}$

Take base-10 logarithms on both sides:

    $\log 0.32=-0.1t$

    $t=-\dfrac{\log 0.32}{0.1}\\\\\\t=-10\log 0.32$

Note that the logarithm of a number decimal number is negative number, thus t will be positive. When you compute that, it is about 4.9 months.