What is the area of this

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 923.558 m²

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, $A_{req}$ ≈ 923.558 m²

Learn more about figures circumscribed by a circle here:

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6 0

2 years ago

Find the first 4 terms in each sequence

koban [17]

Answer:

a₁ = -21 (Given)

a₂ = -28

a₃ = -35

a₄ = -42

Step-by-step explanation:

First term a₁ = -21 (Given)

…………………………………

Second term a₂ :

a₂ = $a_{2-1} - 7\\a_{1} - 7\\a_{1} = -21\\-21 - 7 = -28$

…………………………………

Third term a₃ :

a₃ = $a_{3-1} - 7\\a_{2} - 7\\a_{2} = -28\\-28 - 7 = -35$

…………………………………

Fourth term a₄ :

a₄ = $a_{4-1} - 7\\a_{3} - 7\\a_{3} = -35\\-35 - 7 = -42$

5 0

2 years ago

You must be at least 4 feet tall to go on the ride

Sloan [31]

91038482949394738393858 thank you

5 0

3 years ago

Read 2 more answers

A=1/2 bh for b i need help

qwelly [4]

You said that  A = 1/2 b h

Multiply each side by 2 : 2A = b h

Divide each side by 'h' : 2A/h = b

5 0

3 years ago

Sawyer fills 1/6 of his gas tank in 3/4 of a minute. How many minutes will it take Sawyer to fill his gas tank?

Radda [10]

Answer:

.75 * 6 = 4.5 minutes

Step-by-step explanation:

.75 is 3/4 multiply by 6 because 6/6 is a whole tank of gas.

Sawyers tank takes 4.5 minutes to be filled up.

5 0

2 years ago

Read 2 more answers