Question

If in 3 days the water is 34 and in 19 days its 62 but before UT was 22 what is the equation in standard form

Answer 1

To find the equation in standard form, let's first define what we have.

We have the following data points for the water level:

- \( t_1 = 3 \) days, \( W_1 = 34 \)
- \( t_2 = 19 \) days, \( W_2 = 62 \)

We will first calculate the slope \( m \) of the line that connects these two points using the formula:

\[
m = \frac{W_2 - W_1}{t_2 - t_1} = \frac{62 - 34}{19 - 3}
\]

Calculating the values:

\[
m = \frac{28}{16} = \frac{7}{4}
\]

Now, we have the slope \( m = \frac{7}{4} \).

Next, we can use one of the points (let's use \( (3, 34) \)) to find the equation of the line in point-slope form:

\[
W - W_1 = m(t - t_1)
\]

Substituting the values:

\[
W - 34 = \frac{7}{4}(t - 3)
\]

Now, simplify and express it in slope-intercept form \( W = mt + b \):

\[
W - 34 = \frac{7}{4}t - \frac{21}{4}
\]

Adding 34 to both sides to get \( W \):

\[
W = \frac{7}{4}t + 34 - \frac{21}{4}
\]
\[
W = \frac{7}{4}t + \frac{136 - 21}{4}
\]
\[
W = \frac{7}{4}t + \frac{115}{4}
\]

Next, we want to convert this \( W = \frac{7}{4}t + \frac{115}{4} \) into standard form \( Ax + By = C \).

To eliminate the fraction, multiply through by 4:

\[
4W = 7t + 115
\]

Rearranging it into standard form gives:

\[
-7t + 4W = 115
\]

To fit the convention for standard form, we usually want \( A \) to be positive. So we can multiply the whole equation by -1:

\[
7t - 4W = -115
\]

Thus, the equation in standard form is:

\[
7t - 4W = -115
\]