1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
inna [77]
3 years ago
8

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

the heptagon as shown in the diagram. The area of ΔMSQ is left for a children's playground, and the rest of the garden is planted with flowers. Find the area of the garden planted with flowers.

Mathematics
1 answer:
Alenkinab [10]3 years ago
6 0

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

You might be interested in
Find the area of the semicircle need help asapp
sergij07 [2.7K]

Answer:

2 pi

Step-by-step explanation:

The radius is 2

The area of a circle is

A = pi r^2

We have 1/2 of a circle so

1/2 A = 1/2 pi r^2

         =1/2 pi ( 2)^2

         =1/2 pi *4

      = 2 pi

4 0
3 years ago
A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y)
Alecsey [184]

Answer:

Step-by-step explanation:

The homework  grade, to the nearest integer, for a student with a test grade of 68 is 69.

Step-by-step explanation:

The general form of the linear regression equation is:

Here,

y = dependent variable = test grade

x = independent variable = homework grade

a = intercept

b = slope

5 0
2 years ago
Use the chart to estimate the population density of Romania in 1997. Round to the nearest person.
STALIN [3.7K]

The population density of Romania in 1997 given its area and the population is 95 people per square km.

<h3>What is the population density?</h3>

Population density is the number of people that live in a square kilometer of an area. It is determined by dividing total population by total area.

Population density = population / area

22.600,000 / 237301 = 95 people per square km

To learn more about population density, please check:

brainly.com/question/3704320

#SPJ1

5 0
2 years ago
What is value of this expression below when y=3<br> 8y^2-y-8
serg [7]
Plug in 3 for y
8(3)^2-(3)-8
8(9)-3-8
72-3-8 = 61
8 0
3 years ago
Read 2 more answers
Samantha invests $13,000 at 8% simple interest for 1 year. How much is in the account at the end of the 1 year period?
Nikolay [14]
She will have $14,040 at the end of the year.
4 0
3 years ago
Other questions:
  • This is geometry, I need help! Please and thanks!!
    15·1 answer
  • A can do a piece of work in 4 hours and B can do it in 6 hours. How much time would they take working together to finish the wor
    10·2 answers
  • Can someone help ASAP. Please and thank you!!!
    6·1 answer
  • Explain the steps to finding the vertex of g(x) = 3x2 + 12x + 15
    5·1 answer
  • San Antonio, Texas, is located about 30º north of the equator. If the radius of the Earth is about 3959 miles, how far to the ne
    5·1 answer
  • What is the slope of the line shown below?
    6·1 answer
  • There are over 2700 species of snakes in the world. OVer 600 species are Venemous. Write the fraction of species that are not ve
    10·2 answers
  • Pls help me this is 7th grade (ADVANCED)
    5·1 answer
  • Find the slope of the line (2,1) and (3,1)​
    7·1 answer
  • Five less than 6 times a number plus seven
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!