Solved on Jan 17, 2024

Solve the equation e4t+1et23=10\frac{e^{4 t+1}}{e^{t-2}}-3=10 and select the correct answer.

STEP 1

Assumptions
1. The equation to solve is e4t+1et23=10\frac{e^{4t+1}}{e^{t-2}} - 3 = 10.
2. We assume that tt is a real number.
3. We will use properties of exponents to simplify the equation.

STEP 2

First, we simplify the expression by using the property of exponents that states ea/eb=eabe^{a}/e^{b} = e^{a-b}.
e4t+1et2=e(4t+1)(t2)\frac{e^{4t+1}}{e^{t-2}} = e^{(4t+1)-(t-2)}

STEP 3

Subtract the exponents.
e(4t+1)(t2)=e4t+1t+2e^{(4t+1)-(t-2)} = e^{4t+1-t+2}

STEP 4

Combine like terms in the exponent.
e4t+1t+2=e3t+3e^{4t+1-t+2} = e^{3t+3}

STEP 5

Replace the original fraction in the equation with the simplified expression.
e3t+33=10e^{3t+3} - 3 = 10

STEP 6

Add 3 to both sides of the equation to isolate the exponential term.
e3t+3=13e^{3t+3} = 13

STEP 7

Take the natural logarithm of both sides of the equation to solve for tt. Remember that ln(ex)=x\ln(e^x) = x.
ln(e3t+3)=ln(13)\ln(e^{3t+3}) = \ln(13)

STEP 8

Apply the property of logarithms to simplify the left side of the equation.
3t+3=ln(13)3t+3 = \ln(13)

STEP 9

Subtract 3 from both sides of the equation to isolate the term containing tt.
3t=ln(13)33t = \ln(13) - 3

STEP 10

Divide both sides of the equation by 3 to solve for tt.
t=ln(13)33t = \frac{\ln(13) - 3}{3}

STEP 11

Calculate the value of tt using a calculator.
tln(13)33t \approx \frac{\ln(13) - 3}{3}

STEP 12

Using a calculator, we find that ln(13)2.5649\ln(13) \approx 2.5649.
t2.564933t \approx \frac{2.5649 - 3}{3}

STEP 13

Subtract 3 from 2.5649.
t0.43513t \approx \frac{-0.4351}{3}

STEP 14

Divide by 3 to get the value of tt.
t0.1450t \approx -0.1450

STEP 15

Comparing the calculated value of tt to the multiple-choice options, we find that the closest value is option C t=0.145t=-0.145.
The solution to the equation e4t+1et23=10\frac{e^{4t+1}}{e^{t-2}} - 3 = 10 is t0.145t \approx -0.145, which corresponds to option C.

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