Find the time required for an investment of $5000 to grow to $8000 at an interest rate of 7.5% per year, compounded quarterly.

6.3 years

Step-by-step explanation:

To find the time in years, we will use the Compound interest formula:

        F = P( 1 + i/m)^mn

Where F = future value of investment ($8000); P = Amount invested ($5000); I

i = interest rate (7.5%); m = number of times money is compounded in a year (m = 4 for quarterly) and n = time of investment in years

Substituting;

           8000 = 5000( 1 + 0.075/4)^4n

    Divide both side by 5000 and simplify the bracket on the right hand side;

       8000/5000 = (1.01875)^4n

              1.6 = (1.01875)^4n

Since n is the power, to solve for it we can introduce the natural logarithm   ( ln);

       ln (1.6) = ln (1.01875)^4n

The power can betaken down according to the Laws of logarithms;

     ln (1.6) = 4n x ln(1.01875)

To get n, divide both sides by 4ln (1.01875);

      ln (1.6)/ 4ln(1.01875) = n

  Therefore; n = 6.3 years

first option

a function is a relation in which an input is paired with exactly one output.

so, let's find the right answer.

in our first option, our inputs are january, february, and march. the outputs are 30, 28, and 25. each input has it's own output. this is correct!

our second option we know is incorrect, because ms. valverade has two outputs, dogs and cats. this is not a function.

our third option is incorrect as well. the yellow square has two ouputs, the square and the triangle.

lastly, in our fourth option, we see the number 7 two times with a different output. therefore, this is not a function.

remember: each input must be paired with only one output!