What is the answer to 200e^{-0.01x}=sqrt(200x+49) ?

The answer to 200e^{-0.01x}=sqrt(200x+49) is x=(200\W_{0}(4e^{(0.98)/(200)})-0.98)/4

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200e^{-0.01x}=sqrt(200x+49)

200 e^{- 0.01 x} = 200 x + 49

x = \frac{200 W _{0} ( 4 e ^{\frac{0.98}{200}} ) - 0.98}{4}

Decimal Notation

x = 59.99720 \dots

Solution steps

200 e^{- 0.01 x} = 200 x + 49

Prepare

200 e^{- 0.01 x} = 200 x + 49

for Lambert form:

40000 = (200 x + 49) e^{0.02 x}

Rewrite the equation with

0.02 x + \frac{0.98}{200} = u

and

x = \frac{10000 u - 49}{200}

40000 = (200 (\frac{10000 u - 49}{200}) + 49) e^{0.02 (\frac{10000 u - 49}{200})}

Simplify

40000 = 10000 e^{\frac{0.02 ( 10000 u - 49 )}{200}} u

Rewrite

40000 = 10000 e^{\frac{0.02 ( 10000 u - 49 )}{200}} u

in Lambert form:

e^{u} u = 4 e^{\frac{0.98}{200}}

Solve

e^{u} u = 4 e^{\frac{0.98}{200}} : u = W_{0} (4 e^{\frac{0.98}{200}})

u = W_{0} (4 e^{\frac{0.98}{200}})

Substitute back

u = 0.02 x + \frac{0.98}{200},

solve for

x

Solve

0.02 x + \frac{0.98}{200} = W_{0} (4 e^{\frac{0.98}{200}}) : x = \frac{200 W _{0} ( 4 e ^{\frac{0.98}{200}} ) - 0.98}{4}

x = \frac{200 W _{0} ( 4 e ^{\frac{0.98}{200}} ) - 0.98}{4}

Verify Solutions:

x = \frac{200 W _{0} ( 4 e ^{\frac{0.98}{200}} ) - 0.98}{4}

True

The solution is

x = \frac{200 W _{0} ( 4 e ^{\frac{0.98}{200}} ) - 0.98}{4}

Verify Solutions:

x = \frac{200 W _{0} ( 4 e ^{\frac{0.98}{200}} ) - 0.98}{4}

True

The solution is

x = \frac{200 W _{0} ( 4 e ^{\frac{0.98}{200}} ) - 0.98}{4}

8 = 4^{x + 1}

9^{2 x - 7} = 729

8 = 4^{x + 2}

0 = (\frac{1}{2})^{x} - 4

0 = (\frac{1}{2})^{x} - 3

What is the answer to 200e^{-0.01x}=sqrt(200x+49) ?
The answer to 200e^{-0.01x}=sqrt(200x+49) is x=(200\W_{0}(4e^{(0.98)/(200)})-0.98)/4