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Question 1: 5 pts
Bill wants to plant roses in his triangular plot. There will be 1 plant at a corner. Each row will have 6
additional plants. He wants the plot to have as many lows as possible with 150 rose plants. How many
rows will Bill's plot have?
O 7 rows
O 8 rows
O 6 rows
O 5 rows

Sagot :

The arrangement of the rose plants on the triangular plot is such that they

form a series or progression that is defined.

The number of rows on Bill's plot is; 8 rows

The given parameters for the triangular plot are;

Number of plants at the corner = 1 plant

Number of additional plants per row = 6 plants

Number of rose plants = 150 rose plants

The number of rows in the plot.

The difference between successive rows, d = 6

The number rose at the top vertex, a = 1

Therefore, the rose in the garden forms an arithmetic progression

The first term, a = 1

The common difference, d = 6

The number of rows Bill's plot will have, n is given by the sum of n in terms of

an arithmetic progression, Sn, is given as follows;

[tex]S_n=\frac{n}{2}[2a+(n-1)\times d[/tex]

When Sn = 150, we get;

[tex]150=\frac{n}{2} [2\times 1+(n-1)\times 6[/tex]

150 = 3·n² - 2·n

3·n² - 2·n - 150 = 0

Taking only the positive solution for n, we have;

[tex]n_{1,\:2}=\frac{-\left(-2\right)\pm \sqrt{\left(-2\right)^2-4\cdot \:3\left(-150\right)}}{2\cdot \:3}[/tex]

[tex]n_{1,\:2}=\frac{-\left(-2\right)\pm \:2\sqrt{451}}{2\cdot \:3}[/tex]

[tex]n=\frac{1+\sqrt{451}}{3},\:n=\frac{1-\sqrt{451}}{3}[/tex]

The number of rows Bill's plot has, n ≈ 7.3965

Given that the 7th row is completed, an 8th row will be present on Bill's plot

The number of rows Bill's plot will have = 8 rows

To learn more about the arithmetic progression visit:

https://brainly.com/question/6561461

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