30th term: a = 4, d = 12

A garden is shaped in the form of a regular heptagon (seven-sided), MNSRQPO. A circle with center T and radius 25m circumscribes

Alenkinab [10]

The relationship between the sides MN, MS, and MQ in the given regular heptagon is $\dfrac{1}{MN} = \dfrac{1}{MS} + \dfrac{1}{MQ}$

The area to be planted with flowers is approximately <u>923.558 m²</u>

The reason the above value is correct is as follows;

The known parameters of the garden are;

The radius of the circle that circumscribes the heptagon, r = 25 m

The area left for the children playground = ΔMSQ

Required;

The area of the garden planted with flowers

Solution:

The area of an heptagon, is;

$A = \dfrac{7}{4} \cdot a^2 \cdot cot \left (\dfrac{180 ^{\circ}}{7} \right )$

The interior angle of an heptagon = 128.571°

The length of a side, S, is given as follows;

$\dfrac{s}{sin(180 - 128.571)} = \dfrac{25}{sin \left(\dfrac{128.571}{2} \right)}$

$s = \dfrac{25}{sin \left(\dfrac{128.571}{2} \right)} \times sin(180 - 128.571) \approx 21.69$

$The \ apothem \ a = 25 \times sin \left ( \dfrac{128.571}{2} \right) \approx 22.52$

The area of the heptagon MNSRQPO is therefore;

$A = \dfrac{7}{4} \times 22.52^2 \times cot \left (\dfrac{180 ^{\circ}}{7} \right ) \approx 1,842.94$

$MS = \sqrt{(21.69^2 + 21.69^2 - 2 \times 21.69 \times21.69\times cos(128.571^{\circ})) \approx 43.08$

By sine rule, we have

$\dfrac{21.69}{sin(\angle NSM)} = \dfrac{43.08}{sin(128.571 ^{\circ})}$

$sin(\angle NSM) =\dfrac{21.69}{43.08} \times sin(128.571 ^{\circ})$

$\angle NSM = arcsin \left(\dfrac{21.69}{43.08} \times sin(128.571 ^{\circ}) \right) \approx 23.18^{\circ}$

∠MSQ = 128.571 - 2*23.18 = 82.211

The area of triangle, MSQ, is given as follows;

$Area \ of \Delta MSQ = \dfrac{1}{2} \times 43.08^2 \times sin(82.211^{\circ}) \approx 919.382^{\circ}$

The area of the of the garden plated with flowers, $A_{req}$ , is given as follows;

$A_{req}$ = Area of heptagon MNSRQPO - Area of triangle ΔMSQ

Therefore;

$A_{req}$ = 1,842.94 - 919.382 ≈ 923.558

The area of the of the garden plated with flowers, $A_{req}$ ≈ <u>923.558 m²</u>

Learn more about figures circumscribed by a circle here:

brainly.com/question/16478185

6 0

3 years ago

Which pair(s) of expression are equal in value? Check all that apply

ankoles [38]

Note: Since the symbols for greatest integer function (GIF) and least integer function (LIF) are not given, I will be using the words GIF and LIF in the solution

GIF x means largest integer less than or equal to x, and LIF x means least integer greater than or equal to x.

We need to determine which expressions are equal

Option 1: GIF 4.9 and LIF 3.1

GIF 4.9 = 3

LIF 3.1 = 4

Hence GIF 4.9 ≠ LIF 3.1

Option 2: GIF 15.2 and GIF 14.8

GIF 15.2 = 15

GIF 14.8 = 14

Hence, GIF 15.2 ≠ GIF 14.8

Option 3: GIF -6 and LIF -6

GIF -6 = -6

LIF -6 = -6

Hence, GIF -6 = LIF -6

Option 4: LIF -3.2 and LIF -2.6

LIF -3.2 = -2

LIF -2.6 = -1

Hence, LIF -3.2 ≠ LIF -2.6

Hence, only in Option C the pair of expressions are equal.

3 0

3 years ago

Read 2 more answers

N is prime given that it has 2 digits

rjkz [21]

Minimum 11, Maximum 97

8 0

3 years ago

Which collection of three side lengths will not form a triangle

dezoksy [38]

Answer:

Any collection of lengths (a, b, c) which do not satisfy the triangle inequalities.

Step-by-step explanation:

Any collection (a, b, c) which do not satisfy the triangle inequalities. The inequalities:

a + b > c

b + c > a

a + c > b

You will need to test all of your options on the three inequalities above. If any one of the three fails, the collection won't work.

4 0

3 years ago

Which equation is the inverse of y = 2x^2 - 8?

evablogger [386]

Answer: A

Step-by-step explanation:

$y=2x^2-8$

1. Swap x and y.

$x=2y^2-8$

2. Solve for the new y.

Add 8

$x+8=2y^2$

Divide by 2.

$\frac{x+8}{2}=y^2$

Extract the square root.

$\sqrt{\frac{x+8}{2} }=y$

6 0

3 years ago