P ( t ) = ( 98 - 14 t ^ { 1 / 3 } ) d t
Solve for P
\left\{\begin{matrix}\\P=14\left(-\sqrt[3]{t}+7\right)d\text{, }&\text{unconditionally}\\P\in \mathrm{R}\text{, }&t=0\end{matrix}\right.
Solve for d
\left\{\begin{matrix}d=\frac{P}{14\left(-\sqrt[3]{t}+7\right)}\text{, }&t\neq 343\\d\in \mathrm{R}\text{, }&t=0\text{ or }\left(P=0\text{ and }t=343\right)\end{matrix}\right.
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Pt=\left(98d-14t^{\frac{1}{3}}d\right)t
Use the distributive property to multiply 98-14t^{\frac{1}{3}} by d.
Pt=98dt-14t^{\frac{1}{3}}dt
Use the distributive property to multiply 98d-14t^{\frac{1}{3}}d by t.
Pt=98dt-14t^{\frac{4}{3}}d
To multiply powers of the same base, add their exponents. Add \frac{1}{3} and 1 to get \frac{4}{3}.
tP=98dt-14dt^{\frac{4}{3}}
The equation is in standard form.
\frac{tP}{t}=\frac{14\left(-\sqrt[3]{t}+7\right)dt}{t}
Divide both sides by t.
P=\frac{14\left(-\sqrt[3]{t}+7\right)dt}{t}
Dividing by t undoes the multiplication by t.
P=14\left(-\sqrt[3]{t}+7\right)d
Divide 14td\left(7-\sqrt[3]{t}\right) by t.
Pt=\left(98d-14t^{\frac{1}{3}}d\right)t
Use the distributive property to multiply 98-14t^{\frac{1}{3}} by d.
Pt=98dt-14t^{\frac{1}{3}}dt
Use the distributive property to multiply 98d-14t^{\frac{1}{3}}d by t.
Pt=98dt-14t^{\frac{4}{3}}d
To multiply powers of the same base, add their exponents. Add \frac{1}{3} and 1 to get \frac{4}{3}.
98dt-14t^{\frac{4}{3}}d=Pt
Swap sides so that all variable terms are on the left hand side.
\left(98t-14t^{\frac{4}{3}}\right)d=Pt
Combine all terms containing d.
\frac{\left(98t-14t^{\frac{4}{3}}\right)d}{98t-14t^{\frac{4}{3}}}=\frac{Pt}{98t-14t^{\frac{4}{3}}}
Divide both sides by 98t-14t^{\frac{4}{3}}.
d=\frac{Pt}{98t-14t^{\frac{4}{3}}}
Dividing by 98t-14t^{\frac{4}{3}} undoes the multiplication by 98t-14t^{\frac{4}{3}}.
d=\frac{P}{14\left(-\sqrt[3]{t}+7\right)}
Divide Pt by 98t-14t^{\frac{4}{3}}.
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