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Wes owns a welding shop that focuses on manufacturing tanks to hold salt
water produced from the extraction of oil and gas. His input costs are
\(\$10\thinspace 000.00\) per month, which will produce new tanks and
cover any warranty work required, free of charge to the customer. The
cost of warranty work is equivalent to the profit from two tanks per
month. Wes earns \(\$75\thinspace 000.00\) each month selling the new
tanks. How many tanks must he make and sell each month if he wants to
make a profit of at least \(\$10\thinspace 000.00\) per tank?
Let \(t\) represent the number of tanks made.
Note that Wes uses his supplies for both the new tanks and warranty work, which is equal to two tanks.
The equation for this question will be:
Profit from each tank = earnings from each tank \(-\) cost of producing each tank
LCD: \(t(t + 2)\)
Because Wes cannot make a negative number of tanks, the negative answer can be ruled out as a possibility. Also, since Wes cannot make partial tanks, round the positive answer down to the nearest whole number, \(6\).
Verify for \(t = 6\).
Note that the left side does not equal the right side. However, Wes wanted to make a profit of at least \(\$10\thinspace 000\) per tank, and this verification shows that he will make a profit of \(\$11\thinspace 250\) per tank. In addition, the reason the two sides are not equal is because the rounded value of \(6\) was used in the verification instead of \(6.729\)….
Cost of producing each tank = \(\frac{\$10 \thinspace 000.00}{t + 2}\)
Earnings for each tank = \(\frac{\$75 \thinspace 000.00}{t}\)
Earnings for each tank = \(\frac{\$75 \thinspace 000.00}{t}\)
Note that Wes uses his supplies for both the new tanks and warranty work, which is equal to two tanks.
Profit from each tank = earnings from each tank \(-\) cost of producing each tank
\[10\thinspace 000 = \frac{{75\thinspace 000}}{t} - \frac{{10\thinspace 000}}{{t + 2}}, \thinspace t \ne 0, - 2\]
LCD: \(t(t + 2)\)
\[\begin{align}
\left( {10 \thinspace 000} \right)\left( t \right)\left( {t + 2} \right) &= \left( {\frac{{75\thinspace 000}}{{{\color{red}\cancel{\color{#444}{t}}}}}} \right)\left( {\color{red}{\cancel{\color{#444}{t}}}} \right)\left( {t + 2} \right) - \left( {\frac{{10 \thinspace 000}}{{{\color{red}\cancel{\color{#444}{t + 2}}}}}} \right)\left( t \right) {\color{red}\cancel {\color{#444}{\left(t + 2\right)}}}\\
10\thinspace 000t\left( {t + 2} \right) &= 75\thinspace 000\left( {t + 2} \right) - 10\thinspace 000t \\
10\thinspace 000t^2 + 20\thinspace 000t &= 75\thinspace 000t + 150\thinspace 000 - 10\thinspace 000t \\
10\thinspace 000t^2 - 45\thinspace 000t - 150\thinspace 000 &= 0 \\
5\thinspace 000\left( {2t^2 - 9t - 30} \right) &= 0 \\
\end{align}\]
\left( {10 \thinspace 000} \right)\left( t \right)\left( {t + 2} \right) &= \left( {\frac{{75\thinspace 000}}{{{\color{red}\cancel{\color{#444}{t}}}}}} \right)\left( {\color{red}{\cancel{\color{#444}{t}}}} \right)\left( {t + 2} \right) - \left( {\frac{{10 \thinspace 000}}{{{\color{red}\cancel{\color{#444}{t + 2}}}}}} \right)\left( t \right) {\color{red}\cancel {\color{#444}{\left(t + 2\right)}}}\\
10\thinspace 000t\left( {t + 2} \right) &= 75\thinspace 000\left( {t + 2} \right) - 10\thinspace 000t \\
10\thinspace 000t^2 + 20\thinspace 000t &= 75\thinspace 000t + 150\thinspace 000 - 10\thinspace 000t \\
10\thinspace 000t^2 - 45\thinspace 000t - 150\thinspace 000 &= 0 \\
5\thinspace 000\left( {2t^2 - 9t - 30} \right) &= 0 \\
\end{align}\]
\[\begin{align}
a &= 2,\thinspace b = - 9, \thinspace c = - 30 \\
t &= \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
t &= \frac{{ - \left( { - 9} \right) \pm \sqrt {\left( { - 9} \right)^2 - 4\left( 2 \right)\left( { - 30} \right)} }}{{2\left( 2 \right)}} \\
t &= \frac{{9 \pm \sqrt {321} }}{4} \\
t &= 6.729...\thinspace {\rm{and}} \thinspace t = - 2.229... \\
t &\doteq 6 \thinspace {\rm{and}} \thinspace t \doteq - 2 \\
\end{align}\]
a &= 2,\thinspace b = - 9, \thinspace c = - 30 \\
t &= \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}} \\
t &= \frac{{ - \left( { - 9} \right) \pm \sqrt {\left( { - 9} \right)^2 - 4\left( 2 \right)\left( { - 30} \right)} }}{{2\left( 2 \right)}} \\
t &= \frac{{9 \pm \sqrt {321} }}{4} \\
t &= 6.729...\thinspace {\rm{and}} \thinspace t = - 2.229... \\
t &\doteq 6 \thinspace {\rm{and}} \thinspace t \doteq - 2 \\
\end{align}\]
Because Wes cannot make a negative number of tanks, the negative answer can be ruled out as a possibility. Also, since Wes cannot make partial tanks, round the positive answer down to the nearest whole number, \(6\).
Verify for \(t = 6\).
| Left Side | Right Side |
|---|---|
|
\(10 \thinspace 000\) |
\[\begin{array}{l} \frac{{75\thinspace 000}}{t} - \frac{{10\thinspace 000}}{{t + 2}} \\ \frac{{75\thinspace 000}}{6} - \frac{{10\thinspace 000}}{{6 + 2}} \\ 12\thinspace 500 - 1\thinspace 250 \\ 11\thinspace 250 \\ \end{array}\] |
| \(\hspace{25pt}\)LS \(\ne\) RS | |
Note that the left side does not equal the right side. However, Wes wanted to make a profit of at least \(\$10\thinspace 000\) per tank, and this verification shows that he will make a profit of \(\$11\thinspace 250\) per tank. In addition, the reason the two sides are not equal is because the rounded value of \(6\) was used in the verification instead of \(6.729\)….