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Residents of the town of Maple Grove who are connected to the municipal water supply are billed a fixed amount yearly plus a charge for each cubic foot of water used. A household using $1000$ cubic feet was billed $82$, while one using $1600$ cubic feet was billed $123$.What is the charge in dollars per cubic foot?

Last updated date: 16th Sep 2024
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Hint: Here we will solve this question by forming two linear equations in the form of $ax + by = c$ , where$x$ , and $y$ are two different variables. Then we will compare the equations to find the answer.

Step 1: It is given in the question that the billed amount equals the fixed amount yearly plus a charge for each cubic foot of water used.
Let, variable $x$ equals to fixed amount yearly
And the variable $y$ equals the cubic foot charge of water used. So, the first equation we get:
$\Rightarrow x + 1000y = 82$ ……… (1)
Similarly, the second equation will be:
$\Rightarrow x + 1600y = 123$ ……… (2)
Step 2: By subtracting equation (1) from equation (2), we get:
${\text{ }}x + 1600y = 123 \\ \pm x \pm 1000y = \pm 82 \\ \;\overline {0x + 600y = 41} \\$
OR $\Rightarrow 600y = 41$
By taking$600$ into the RHS side, we get:
$\Rightarrow y = \dfrac{{41}}{{600}}$
By dividing the RHS side we get the value of charge in dollars per cubic foot:
$\Rightarrow y = \ 0.06$

Charge in dollars per cubic foot is $\ 0.06$.

Note: We can also solve these types of problems by a graphical method with two different variables $x$ that denote cubic foot of water used and $y$which denotes the total price paid, we have two values of $x$ i.e.
${x_1}$ , ${x_2}$ and similarly for $y$ also i.e. ${y_1}$ , ${y_2}$.
By using the formula of the slope $m$ which defines the change in $y$ over the change in $x$ of a line:
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
By substituting the values of ${x_1} = 1000$ , ${x_2} = 1600$, ${y_1} = 82$ and ${y_2} = 123$ in the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ , we get:
$\Rightarrow m = \dfrac{{123 - 82}}{{1600 - 1000}}$
By subtracting the numerator and denominator terms in the equation $m = \dfrac{{123 - 82}}{{1600 - 1000}}$, we get:
$\Rightarrow m = \dfrac{{41}}{{600}}$
By dividing the RHS side we get the value of charge in dollars per cubic foot:
$\Rightarrow m = \ 0.06$