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Case Study - 1
In a garden, saplings of rose flowers were planted at equal intervals to form a spiral pattern. The spiral is made up of successive semicircles, with centres alternatively at A and B, starting with centre at A, of radii $50\text{ cm}, 100\text{ cm}, 150\text{ cm}, \dots$ as shown in the figure given below. Spiral 1 has 10 flowers, Spiral 2 has 20 flowers, Spiral 3 has 30 flowers and so on.
Based on the above information, answer the following questions :
(i) What is the radius of the $13^{th}$ spiral ?
(ii) If the radius of the $n^{th}$ spiral is $500\text{ cm}$, find the value of $n$.
(iii) (a) Find the total number of saplings till the $11^{th}$ spiral.
OR
(b) Till which spiral, will there be a total of 450 saplings ?
In a garden, saplings of rose flowers were planted at equal intervals to form a spiral pattern. The spiral is made up of successive semicircles, with centres alternatively at A and B, starting with centre at A, of radii $50\text{ cm}, 100\text{ cm}, 150\text{ cm}, \dots$ as shown in the figure given below. Spiral 1 has 10 flowers, Spiral 2 has 20 flowers, Spiral 3 has 30 flowers and so on.
Based on the above information, answer the following questions :
(i) What is the radius of the $13^{th}$ spiral ?
(ii) If the radius of the $n^{th}$ spiral is $500\text{ cm}$, find the value of $n$.
(iii) (a) Find the total number of saplings till the $11^{th}$ spiral.
OR
(b) Till which spiral, will there be a total of 450 saplings ?
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Solution: (i) $a_{13} = 650\text{ cm}$
(ii) $a_n = 500$
$50 + (n - 1)50 = 500$
$n = 10$
(iii) (a) $a = 10, d = 10$
$S_{11} = \frac{11}{2} [20 + 10 \times 10] = 660$
OR
(b) $a = 10, d = 10$
$450 = \frac{n}{2} [20 + (n - 1) 10]$
$n^2 + n - 90 = 0$
$n = 9$
(ii) $a_n = 500$
$50 + (n - 1)50 = 500$
$n = 10$
(iii) (a) $a = 10, d = 10$
$S_{11} = \frac{11}{2} [20 + 10 \times 10] = 660$
OR
(b) $a = 10, d = 10$
$450 = \frac{n}{2} [20 + (n - 1) 10]$
$n^2 + n - 90 = 0$
$n = 9$