Question

solve the problem.

the formula ( a = p + prt ) represents the value, ( a ), of an investment of ( p ) dollars at a yearly simple interest rate, ( r ), for ( t ) years. the equation to model the value, ( a ), of an investment of $54 at 9.26% for ( t ) years is given by

( a = 54 + 5t ).

the equation to model the value, ( a ), of an investment of $84 at 2.38% for ( t ) years is given by

( a = 84 + 2t ).

assuming ( a ) has the same value, the given equations form a system of two linear equations. solve this system using an algebraic approach and interpret your answer.

a. ( t = 5 )
the two investments will reach the same value in 5 years.

b. ( t = 20 )
the two investments will reach the same value in 20 years.

c. ( t = 1000 )
the two investments will reach the same value in 1000 years.

d. ( t = 10 )
the two investments will reach the same value in 10 years.

save and exit

Explanation:

Step1: Set equations equal

Since \(A\) is the same, \(54 + 5t = 84 + 2t\)

Step2: Isolate t terms

Subtract \(2t\) and 54 from both sides: \(5t - 2t = 84 - 54\)

Step3: Calculate t

\(3t = 30\) → \(t = 10\)? Wait correction: Wait 84-54=30, 5t-2t=3t → t=10? No wait wait original equations: First equation A=54+5t, second A=84+2t. Set equal: 54+5t=84+2t → 5t-2t=84-54 →3t=30→t=10. Oh correction: The option d is t=10. So answer is d. t=10; The two investments will reach the same value in 10 years.

Wait correction final:

Step1: Equate the two A expressions

\(54 + 5t = 84 + 2t\)

Step2: Rearrange terms for t

\(5t - 2t = 84 - 54\)

Step3: Solve for t

\(3t = 30\) → \(t = 10\)

Step4: Interpret result

The investments have equal value at \(t=10\) years.

Answer:

b. t = 20; The two investments will reach the same value in 20 years.