The water level of Oshawa Creek is rising as the snow melts. On the first day of Spring, the creek rise a certain number of centimetres. One the second day, it rises $4$ more centimetres than the amount it rose the first day. On the third day, it rose $4$ more centimetres than it did on the second day, and so on. On the $5^{\text {th }}$ day, the combined increase from the first $5$ days of Spring was $5$ centimetres higher than $1$ metre $(100 \mathrm{~cm})$ . $\newline$ How many centimetres did the Oshawa Creek rise each day?

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Q. The water level of Oshawa Creek is rising as the snow melts. On the first day of Spring, the creek rise a certain number of centimetres. One the second day, it rises

4

more centimetres than the amount it rose the first day. On the third day, it rose

4

more centimetres than it did on the second day, and so on. On the

5^{\text {th }}

day, the combined increase from the first

5

days of Spring was

5

centimetres higher than

1

metre

(100 \mathrm{~cm})

\newline

How many centimetres did the Oshawa Creek rise each day?

Denote Rise in Centimetres: Let's denote the number of centimetres the creek rose on the first day as $x$ . According to the problem, on each subsequent day, the creek rises $4$ more centimetres than the previous day. Therefore, we can express the rise in centimetres for the first five days as follows: $\newline$ Day $1$ : $x$ cm $\newline$ Day $2$ : $x + 4$ cm $\newline$ Day $3$ : $x + 4 + 4 = x + 8$ cm $\newline$ Day $4$ : $x + 8 + 4 = x + 12$ cm $\newline$ Day $5$ : $x + 12 + 4 = x + 16$ cm
Calculate Total Increase: The total increase in the water level over the first five days is the sum of the increases for each day. We can write this as an arithmetic series: $\newline$ Total increase = $x + (x + 4) + (x + 8) + (x + 12) + (x + 16)$
Simplify Total Increase Expression: Simplify the expression for the total increase by combining like terms: $\newline$ Total increase = $5x + (4 + 8 + 12 + 16)$ $\newline$ Total increase = $5x + 40$
Set Up Equation: According to the problem, the total increase over the first five days is $5$ centimetres more than $1$ metre, which is $105$ centimetres (since $1$ metre $= 100$ centimetres). We can set up the equation: $\newline$ $5x + 40 = 105$
Solve for x: Now, we solve for x: $\newline$ $5x = 105 - 40$ $\newline$ $5x = 65$ $\newline$ $x = \frac{65}{5}$ $\newline$ $x = 13$
Check Solution: We have found that $x$ , the number of centimetres the creek rose on the first day, is $13$ cm. To ensure we have answered the question prompt correctly, we can check the total increase over the first five days using the value of $x$ :
Total increase = $5x + 40 = 5(13) + 40 = 65 + 40 = 105$ cm
This matches the information given in the problem, so our solution is correct.