Martha Stewart wants to rebuild her ugly garden. She thinks it needs a modern twist and decides to make a star-shaped garden where the legs of the star will be filled with flowers and the central pentagon will be grass.
Martha needs your help to calculate the floral area and grassy area in this new garden design. Then she can figure out the equipment and materials she will need for the project.
To form this star, simply extend the sides of the pentagon to make the legs of the star. Here are the dimensions of her stellar garden:
each leg of the star is a triangle with two sides, both 20 feet long, and an height of 19 feet.
The pentagon is regular (the sides are all equal and the angles are all equal),
and the distance from the center of the pentagon to one of the sides is 8.5 feet.
To the nearest square foot, what are the areas of Martha's flower section and grass section. Because Martha is not sure what she would like to do, please provide three different ways she could arrange her garden, be sure to let her know the area each floral and grass area will take. Explain how you solved one of the area problems. BE SPECIFIC! Walk us through all of your steps.
HERE ARE SOME OF YOUR SOLUTIONS:
8 comments:
RW
13
12
13
13
25
4
2636400
To find out how many different ways there is to do what the steps say u need to multiply how many of each one with the other
13 spades times the 12 spades left over
Times 13 clubs times the 13 diamonds,
Times 25(there are 13 hearts left and 12 diamonds)
And times the 4 aces l
Equaling 2636400
P.O.W.
C.H.
The Arms:
You can use the Pythagorean theorem to find the area of the triangles by splitting them in half. It gives “A” and “C”. You do 19 squared and then 20 squared. Then subtract 361 from 400 which gives you 39 which “B”. Multiply 6.24 by 19 to get 118.56. Multiply this by 5 to get 592.8. This is the area of all five triangles in the star.
The Pentagon: The pentagon can be split up into five different triangles. Each side of the pentagon is 12.48 feet long, making the base of each triangle 12.84 and the height 8.5.
Multiply 12.48 by 8.5 to get 106.08. Multiply this by 5 to get the area of the pentagon which is 530.4.
The total area of the star is 1123.2 square feet.
530.4 square feet of grass and 592.8 square feet of flowers.
Extra way #1: You could rearrange the flowers and the grass one way by putting flowers where the grass is and grass where the flowers are.
The area of the flowers would be 530.4 square feet. The area of the grass would be 592.8 square feet.
Extra way #2: Another way to rearrange the flowers and grass is by having the flowers and grass alternating on the stars arms, with two arms with grass and three with flowers and flowers in the middle. The area of all the flowers is 848.64 square feet. The area of all the grass is 742.56 square feet.
The area of each of the triangles is 118.75 . I divided each isosceles triangle into 2 right triangles. The one leg is 19 feet and the hypotenuse is 20 feet. We tried to do the Pythagorean Theorem backwards. The base of the right triangle is 6.25. 6.25 x 19=118.75 so 118.75 x 5 legs =593.75 So, the area of all five legs of the star is 593.75. That’s how much area Martha Stewart can put flowers in.
I have no idea how to find the area of the pentagon. I tried but did not succeed. 12.5 x 5 is 62.5 and that is the perimeter which I don’t know what to do with.
Actually, Johnny just helped me so he says that the
perimeter x 3 so maybe it is 187.5????????
P.O.W. February 1, 2008 J.K.
Here the problem gives us a pentagon with extended sides to forms triangles. This is a very strange problem due to the fact that I’m not sure how to find the area of a pentagon. However, I will try my best. 265.41
I used the Pythagorean Theorem to find the length of DB. I actually reversed the Theorem. 20 is the length of the Hypotenuse. That is also C. In order to find C squared you must square 20 to get 400. That is C squared. The formula is A sq. plus B sq. equals C sq. A=19 and B=??. If you square 19 you get 361. Now 361 plus B sq. =400. 400-361=39 or B sq. Now you must unsquare 39 or find the square root of 39. Radical 39=about 12.49. So, 12.49 sq. +19sq. =20 sq. Now that you have the base, 12.49, of 1 triangle which is DB, there is one thing you must also know, the formula for finding the area of a triangle. Base times height divided by 2. The base is 12.49 and the height is 19, so 12.49 times 19=237.31 now you must divide that by 2. 237.31 divided by 2=118.655. 118.655 is the area of 1 of the triangle arms. There are 5 arms so you must multiply 118.655, which is the area of one triangle, by 5. The answer is 593.275.
So, in conclusion for the total area where Martha Stewart can place flowers
is 593.275 sq. ft.
Now we need to find the area of the pentagon, where Martha will plant her grass. From the center of the Pentagon to d to b forms a triangle. Now if you rotate the triangle from the center point it fits into triangle 4 more times for a total of 5 times. Formula for area of a triangle is .5base*height so 12.49 times 8.5=106.165. Now multiply that by one half and you will get 53.08. That is the area of triangle ABD. Now multiply that by 5 for the whole pentagon and you will get 265.4125.
The total area where Martha Stewart will plant her grass is 265.41 sq. ft.
For the whole area of the star the area is 858.6875 sq. ft.
D.E.
Star Bright Star Light
Well, to find the area, you have to divide the star into different parts. I start with the triangles. I split the triangles in half, and make them right triangles to find the area of them. A right triangle has a length of 19 and a hypotenuse of 20. So I do 20x20 which is 400, and I do 19x19 which is 361, and I take away 400 from 361 and get 39. Then I use a calculator and get the square root of 39. I get 6.244 that is the width. Then, 6.244 multiplied by 19 becomes the area, and then that multiplied by 2 is the whole triangle. So 19 times 6.244 are 118.63, then that divided by 2 is 59.5. Now because all the pentagon sides are the same, and all the triangles have the same hypotenuses and lengths, you find the area of all five triangles by multiplying 118.63 by 5 for all the triangles is 593.18 sq ft. So now I’ve gotten the triangles out of the way, now I just need the area of the pentagon. You find the area of a pentagon by splitting it up into different parts, and I split them up into right triangles. Then I get 265 by multiplying 8.5 by 6.244, then dividing by two to get 53.5, so I multiply that by 5 and get the pentagon’s area: 265. 265+591.8 is 857.17 sq ft, the whole area of the star.
Then to find three different arrangements, one arrangement is when you put triangles KJB and EDF on sides HG and HI so that it will look like an arrow. Another arrangement is taking triangles FGH and HJI and place on the sides of BK and ED, and it will look like a jester. The last one is taking triangles FGH and HJI then putting them on each side of triangle CDB, making a trapezoid over the guy’s head.
R.B.
Star Light Star Bright
WORK
Area of a triangle (red and blue): 118.655 square units
Area of one right –angle triangle in pentagon: 26.54 square feet
There are 10 right-angle triangles (area of pentagon): 265.4 square feet
ANSWER
The area of the pentagon is 265 square feet (rounded). I found this by using the bottom leg of the triangle and divide it by 2. Then I used the paint and made 10 triangles with legs of 6.245 and 8.5. Then I did that area times 10 and got 265.4 and rounded is 265 square feet. The area of one triangle is 119 square feet (rounded). I found this area by using the base in half and timed that by the height of 19 and found the area of 118.655 which rounded is 118.655.
One extra way to organize the shape is putting the triangles on the vertices of the pentagon. Another way is putting and flipping the triangle BJK and fitting it by line HI and then move and flip triangle EFD and putting it by line HG. The last way is putting triangle EFD on line CD and putting triangle BJK on line CB.
C.C.
Pentagon Area: 62 feet
Arms of Star Area: 617.5 feet
1. Martha can arrange her garden by making the stars legs then the pentagon would fit inside the triangle legs
2. Martha can create her pentagon then find the height of each leg where markers can be placed can be placed then triangles can be made.
3. Martha can create her star by making smaller triangles in her pentagon. Then the legs of the star can be found.
3. I found the area of the pentagon and legs by first making five triangles out of the pentagon. I did this by finding the length from the middle of the pentagon to the side. After finding the area of the pentagon you can find the height and mark the height of the triangle. Once finished you fill the star in with grass and flowers.
E.K.
The area of each of the triangles is 118.75 . I divided each isosceles triangle into 2 right triangles. The one leg is 19 feet and the hypotenuse is 20 feet. We tried to do the Pythagorean Theorem backwards. The base of the right triangle is 6.25. 6.25 x 19=118.75 so 118.75 x 5 legs =593.75 So, the area of all five legs of the star is 593.75. That’s how much area Martha Stewart can put flowers in.
I have no idea how to find the area of the pentagon. I tried but did not succeed. 12.5 x 5 is 62.5 and that is the perimeter which I don’t know what to do with.
Actually, Johnny just helped me so he says that the
perimeter x 3 so maybe it is 187.5????????
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