Answer :
To solve the equation [tex]101325 \cdot 10^{-0.054 \cdot h} = 60000[/tex], we are looking for the value of [tex]h[/tex]. This is an exponential equation that can be solved by following these steps:
Isolate the Exponential Part:
Start by dividing both sides of the equation by 101325 to isolate the exponential expression:
[tex]10^{-0.054 \cdot h} = \frac{60000}{101325}.[/tex]
Calculate the right side to simplify:
[tex]10^{-0.054 \cdot h} \approx 0.591.[/tex]
Take the Logarithm of Both Sides:
To solve for [tex]h[/tex], take the logarithm of both sides. You can use the base-10 logarithm (common logarithm) since the base of the exponent is 10:
[tex]\log_{10}(10^{-0.054 \cdot h}) = \log_{10}(0.591).[/tex]
Simplify Using Logarithmic Properties:
Apply the power rule for logarithms, which states that [tex]\log_{10}(10^x) = x[/tex]:
[tex]-0.054 \cdot h = \log_{10}(0.591).[/tex]
Calculate the [tex]\log_{10}(0.591)[/tex] using a calculator:
[tex]\log_{10}(0.591) \approx -0.228.[/tex]
Solve for [tex]h[/tex]:
Divide both sides by [tex]-0.054[/tex] to solve for [tex]h[/tex]:
[tex]h = \frac{-0.228}{-0.054}.[/tex]
Compute the division:
[tex]h \approx 4.222.[/tex]
Therefore, the solution to the equation is [tex]h \approx 4.222[/tex]. This means that for [tex]h[/tex] approximately equal to 4.222, the left-hand side of the original equation equals 60000.
