Answer:
After 7.04 years the amount will reach $57,000 or more
Step-by-step explanation:
The rule of the compound interest is [tex]A=P(1+\frac{r}{n})^{nt}[/tex] , where
∵ A loan of $36,000 is made at 6.75% interest, compounded annually
∴ P = 36,000
∴ r = 6.75% = 6.75 ÷ 100 = 0.0675
∴ n = 1 ⇒ compounded annually
∵ The amount after t years will reach $57,000 or more
∴ A = 57,000
→ To find t substitute these values in the rule above
∵ 57,000 = 36,000 [tex](1+\frac{0.0675}{1})^{(1)(t)}[/tex]
∴ 57,000 = 36,000 [tex](1.0675)^{t}[/tex]
→ Divide both sides by 36,000
∵ [tex]\frac{19}{12}[/tex] = [tex](1.0675)^{t}[/tex]
→ Insert ㏒ in both sides
∴ ㏒( [tex]\frac{19}{12}[/tex] ) = ㏒ [tex](1.0675)^{t}[/tex]
→ Remember ㏒[tex]a^{n}[/tex] = n ㏒([tex]a[/tex])
∵ ㏒( [tex]\frac{19}{12}[/tex] ) = t ㏒(1.0675)
→ Divide both sides by ㏒(1.0675)
∴ 7.035151337 = t
∴ t ≅ 7.04
∴ After 7.04 years the amount will reach $57,000 or more
