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Popular >

8=60x-1800+(1800)/(e^{x/(30))}

Solution

8 = 60 x - 1800 + \frac{1800}{e ^{\frac{x}{30}}}

Solution

x = 30 W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}, x = 30 W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

+1

Decimal Notation

x = - 2.78467 \dots, x = 2.87357 \dots

Solution steps

8 = 60 x - 1800 + \frac{1800}{e ^{\frac{x}{30}}}

Prepare

8 = 60 x - 1800 + \frac{1800}{e ^{\frac{x}{30}}}

for Lambert form:

1800 = (- 60 x + 1808) e^{\frac{x}{30}}

Rewrite the equation with

\frac{1}{30} x - \frac{226}{225} = u

and

x = 30 u + \frac{452}{15}

1800 = (- 60 (30 u + \frac{452}{15}) + 1808) e^{\frac{( 30 u + \frac{452}{15} )}{30}}

Simplify

1800 = - 1800 e^{\frac{225 u + 226}{225}} u

Rewrite

1800 = - 1800 e^{\frac{225 u + 226}{225}} u

in Lambert form:

e^{u} u = - \frac{1}{e ^{\frac{226}{225}}}

Solve

e^{u} u = - \frac{1}{e ^{\frac{226}{225}}} : u = W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}), u = W_{0} (- \frac{1}{e ^{\frac{226}{225}}})

u = W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}), u = W_{0} (- \frac{1}{e ^{\frac{226}{225}}})

Substitute back

u = \frac{1}{30} x - \frac{226}{225},

solve for

x

Solve

\frac{1}{30} x - \frac{226}{225} = W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) : x = 30 W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

Solve

\frac{1}{30} x - \frac{226}{225} = W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) : x = 30 W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

x = 30 W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}, x = 30 W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

Verify Solutions:

x = 30 W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

True

, x = 30 W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

True

The solutions are

x = 30 W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}, x = 30 W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

Verify Solutions:

x = 30 W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

True

, x = 30 W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

True

The solutions are

x = 30 W_{- 1} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}, x = 30 W_{0} (- \frac{1}{e ^{\frac{226}{225}}}) + \frac{452}{15}

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