Can you find area of the Green shaded Rectangle? | (Two Methods) |
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- Опубликовано: 22 окт 2024
- Learn how to find the area of the Green shaded Rectangle. Important Geometry and Algebra skills are also explained: rectangles. Step-by-step tutorial by PreMath.com
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Can you find area of the Green shaded Rectangle? | (Two Methods) | #math #maths | #geometry
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First comment and first like, can you pin it?
You got it!😀
Thank you sir
With method 2, a variant once you've got the equations for the areas of the yellow and blue rectangles would be to multiply the first equation by 7 and the other by 11. You'd get 77h - 7xh = 217, and 77h - 11xh = 187. Subtract the second from the first to get 4xh = 30. Divide by 4 to get xh = 15/2.
The question was to find the area of the green rectangle, not necessarily all of the dimensions.
Thanks for the feedback ❤️
Excellent
Let x is width ; y is area of green rectangle
Area of green and yellow rectangle.
7x=17+y (1)
Area of green and Blue rectangle
11x=31+y (2)
(1) y=7x-17
(2) 11x=31+7x-17
11x-7x=31-17
4x=14
so x=7/2
(1) y=7(7/2)-17=15/2cm^2=7.5cm^2❤❤❤ thanks sir.
Excellent!
You are very welcome!
Thanks for sharing ❤️
Partition the blue rectangle in such a way, you divide it into two more rectangles. A yellow rectangle with an area of 17 cm² and a blue rectangle with an area of 14 cm². Together, the rectangles have a total area of 31 cm², and the new yellow rectangle is beside the green one.
Therefore, the distance between the bottom left vertex of the green rectangle and the bottom right vertex of the right yellow rectangle is 7 cm.
So, the length of the small blue rectangle is 11 - 7 = 4 cm.
A = lw
14 = 4w
w = 14/4
= 7/2 = 3.5
So, the rectangles have a width of 3.5 cm.
Find the total area of the left yellow rectangle and the green rectangle combined.
A + 17 = 7 * 3.5
A + 17 = 24.5
A = 7.5
So, the area of the green rectangle is 15/2 square centimeters, or 7.5 square centimeters.
Excellent!
Thanks for sharing ❤️
Let's find the area:
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Since all rectangles have the same height, we can conclude:
[A(yellow) + A(green)]/[width(yellow) + width(green)] = [A(blue) + A(green)]/[width(blue) + width(green)]
[17cm² + A(green)]/7 = [31cm² + A(green)]/11
187cm² + 11*A(green) = 217cm² + 7*A(green)
4*A(green) = 30cm²
⇒ A(green) = 7.5cm²
Best regards from Germany
Wunderbar !
Excellent!
Thanks for sharing ❤️
That's an answer to the problem! Thanks for sharing!
Height = h, Yellow area = y, Green area = g, Blue area = b.
11h = 31 + g.
7h = 17 + g.
Subtracting, 4h = 14.
h = 14 / 4 = 7 / 2.
y + g = 7h = 49 / 2 = 24.5.
y = 17.
Therefore g = 24.5 - 17 = 7.5.
Excellent!
Thanks for sharing ❤️
7h - 17=11h-31 , 4h=14 , h=7/2 . X= 7x7/2-17=49/2-17=24,5-17=7,5 .
label the horizontal side of the green rectangle a and h the height of the rectangles
area of the blue rectangle = (11-a)h=31
11h-ah=31 (eqtion $)
area of the green rectangle (7-a)h=17
7h-ah=17 (equation &)
subtract the two equations to get
4h=14
h=3.5
now add them to get 18h-2ah=48
9h-ah=24
ah=9h-24
ah=9*3.5-24
ah=31.5-24=7.5
the area of the green region is a*h=7.5 cm^2
Here's an easier one
*3rd Method*
17+A = 7h
31+A= 11h
A is the area of the center rectangle and h is the height
Multiply either equations by -1
We get
4h= 14
h= 7/2
Then solve for A
7(7/2) = 17+A
49/2= 34/2+A
15/2 = A
Green rectangle area : X height of the rectangle : h
7h=17+X 11h=31+X
X=7h-17 X=11h-31 7h-17=11h-31 4h=14 h=3.5 X=7.5
Green Rectangle area : 7.5cm²
Excellent!
Thanks for sharing ❤️
Altura =h=(31-17)/(11-7)=7/2.
Base amarilla =a =17/(7/2)=34/7 ; base verde =v =7-(34/7)=15/7.
Área verde =v*h=(15/7)*(7/2)=15/2=7,5.
Gracias y saludos.
I love your channel, it has taught me so many patterns that has now allowed me to solve such problems very quickly!
Wonderful!
Glad to hear that!
Thanks for the feedback ❤️
x = green area
y = height of the rectangle(s)
A = green + yellow areas
B = green + blue areas
A = 7y ---> [1]
B = 11y ---> [2]
A = x + 17 ==> x = A - 17 ---> [3]
B = x + 31 ==> x = B - 31 ---> [4]
First method:
A = 7y
Another super clear demonstration Professor ! Thanks for sharing !
Glad you liked it!
Thanks for the feedback ❤️
Method 3: Let the unknown area = x and the height = h, then the area of {17+x} = 7*h and the area of {31+x} = 11*h.
Two simultaneous equations; just solve for x. Subtract and solve for h first e.g. x-x=0, and then sub h into either of the equations.
Let the Games commence!!
In first place i would like to say that this a sort of a Logic Problem : P(A U B) = P(A) + P(B) - P(A ∩ B)
1) Let's call the Height of this Rectangle "h" : h = k (constant)
2) Let's cal the Area of the Green Rectangle "X"
What do we got?
3) The Total Area of the Tricolor Rectangle is : (17 + X + 31) = (48 + X) sq cm
4) Base of Yellow Triangle is "a cm"
5) Base of Green Triangle is "b cm"
6 Base of Blue Triangle is "c cm"
So :
7) a * h = 17 sq cm
8) b * h = X sq cm
9) c * h = 31 sq cm
Now the best part!!
10) a + b = 7 cm
11) b + c = 11 cm
12) (a + b) * h = 17 + X ; 7 * h = (17 + X) ; h = (17 + X) / 7
13) (b + c) * h = 31 + X ; 11 * h = (31 + X) ; h = (31 + X) / 11
14) Equivalence Principle; If : x = y and x = z ; then : y = z
15) ((17 + X) / 7) = ((31 + X) / 11) ; 11 * (17 + X) = 7 * (31 + X) ; 187 + 11X = 217 + 7X ; 217 - 187 = 11X - 7X ; 30 = 4X ; X = 30 / 4 ; X = 15 / 2 sq cm X = 7,5 sq cm
16) My Final Answer is that the Green Rectangle Area is equal to 7,5 Square Centimeters.
This is the Beauty of Mathematics!!
Excellent!
Thanks for sharing ❤️
Thank you i solved it 😌 ❤ it was tasty 😋 😜
Great 👍
You are very welcome!
Thanks for the feedback ❤️
Rather easy, (17+g)/(g+31)=7/11, 11×17+11g=7g+7×31, 4g=7×31-11×17=30, g=15/2😊.
Great solution!
Very nice solution!
Glad you think so!
Thanks for the feedback ❤️
I used the "k" method. k is the width for the "17" region. 31/17k is the width for the "31" region. 31/17k + x = 11 and k + x = 7. Solve for k and x which are 34/7 and 15/7 respectively. h being the height can be found by solving for h in the equation 31 = 62/7*h so h = 7.5. I know "Dr. Premath" likes this method and uses it on other problems. I was surprised he didn't have it in his usual array of solutions.
I haven't calculated the width (x) of the unknown rectangle, only the height (h).
From the second method, I started here from the calculation below (about 8:15 in the video)
7h - xh = 17 --> xh = 7h - 17
11h - xh = 31 --> xh = 11h - 31
7h - 17 = 11h - 31
4h = 14 --> h = 3.5
Now you can calculate the unknown green area with one of the formulas above
xh = 7h - 17 = 7.5
xh = 11h - 31 = 7.5
I don't pretend my approach is better, it's just a slightly different one.
I solve like this:
The base of the rectangle whose area is 17 cm^2, I called "a". The base of the rectangle whose area is unknown, I called "b". And the base of the rectangle whose area is 31 cm^2, I called "c". And their height is the same and I called it "h".
Then:
a+b = 7 ==> 1
c+b = 11==> 2
2 - 1
c - a = 11 - 7
c - a. = 4
c = a + 4
And using Areas:
31 = h*c
31 = h(a - 4)
31 = h*a - 4h ==> 3
17 = h*a ====> 4
3 - 4
31 - 17 = 4h
4h = 14
h = 14/4
h = 7/2
And:
h*a = 17
(7/2)*a = 17
7a = 34
a = 34/7
a + b = 7
b = 7 - (34/7)
b = (49/7) - (34/7)
b = 15/7
Area = b*h
Area = (15/7)*(7/2)
Area = 15/2
Area = 7,5 cm^2
I think the answer is arrived (step 2) at 9:35 (pause and verify).. the moment we have 7/2 X ( which is length and breath of the green rectangle) the answer is done. No need to find the side of the green triangle ?
Thanks for the feedback ❤️
Very good. The first method wouldn't have occurred to me. Thanks.
My solution was a variant of the 2nd method, solving directly for A rather than via x.
A = 7h - 17
A = 11h - 31
h = 3.5
A = 7.5
I preferred the 2nd method. IMO it was more simple and easy. Especially if we don't bother to find the width and length of the green rectangle, just the (width X length).
Excellent!
Thanks for the feedback ❤️
Thanks Sir
That’s very nice
With glades
17+A=7y
31+A=11y
14=4y
y=7/2
17+A=49/2
A=49/2-34/2
A=15/2
best approach I've seen here...
Thank you!
You don't need to find the value of x. Just substitute y in one of the equations and you will find the value of xh, which is precisely the area of the green rectangle.
Easiest method (im my opinion):
11h = A + 31
7h = A + 17
4h = 14
h = 3.5
A + 17 = 7 * 3.5 = 24.5
A = 24.5 - 17 = 7.5 square units
Excellent!
Thanks for sharing ❤️
Similar but simpler
11h = A + 31
7h = A+ 17
77h = 7A + 217
77h = 11A+ 187
Subtract
0 = 4A - 30
4A = 30
A = 30/4 = 7.5
@@bubba208 I know what you mean by "simpler": more direct to the solution, no need to calculate h first. 👍
In my opinion, the calculations are a bit more difficult to do in your head. Thanks for sharing.
I guess the easiest method by far is the one that I found in some other comments:
(A + 17) : 7 = (A + 31) : 11
11 (A + 17) = 7 (A + 31)
11A + 187 = 7A + 217
4A = 30
A = 7.5 [cm²]
3rd method X=area of green zone , (31+X)/(17+X)=11/7 therefore 217+7X=187+11X or 30=4X or X=7.5
I used a third method. I set the problem like method two, but noticed both equations had a hx (area) term. I multiplied equation 1 by 11 and equation 2 by 7, then subtracted. Then solved for hx. This saved a few steps.
S=7,5 cm²
Excellent!
Thanks for sharing ❤️
(S+31)/(S+17)=11/7. So, S=7.5 square units.
Great solution!
(17+S) : (31+S) = 7 : 11
S = 7.5
I used the second approach.
It took less than 3 min
7.5 Sq.units
Excellent!
Thanks for sharing ❤️
TGIF! 🙂
Yessss!
Have a good weekend😀
Area= 7.5 sq cm😅
Excellent!
Thanks for sharing ❤️