I am volunteering some of my time to help students understand some of the more difficult topics in school. In doing so, I hope that I can help make school much easier.
Question Rectangular garden is 85 meter long and 57 meter broad. Trees are to be planted around the garden at a distance of 5 meter having a distance of 4 meter the start. How many such trees can be planted
Solution Procedure 1
compute for the perimeter of the rectangle that is available for tree planting
L: 85 meter - 4 meter = 81 meter
W: 57 meter 4 meter = 53 meter Procedure 2
The perimeter of a rectangle is computed by 2L + 2W
2 (81 meter) + 2 (53 meter)
162 meter + 106 meter = 268 meter Procedure 3
How many meters can be planted?
268 meter / 5 meter = 53 trees
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Question A pond is enclosed by a wooden deck that is 3 feet wide. The fence surrounding the deck is 100 feet long. If the pond is rectangular and the length of the pond is to be three times its width, what are its dimensions?
Solution Let x = width of the pond 3x = length of the pond Perimeter of the rectangle = 100 ft The formula for perimeter is 2x + 6x = 100
Hence, 8x = 100 x = 12.5
Therefore, the width of the pond is 12.50 feet and its length is 37.50 feet.
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Question If the perimeter of a rectangular block of land is 46 meters, find the lenght and width of the block.
Solution The formula for a rectangle's perimeter is 2L + 2W where L is length and W is width. Hence the above problem can be written as 2L + 2W = 46 which is also equal to L + W = 23, however it can not be solved yet given the information. The relationship of the rectangle's width and length must be stated in rder for the problem to be solved.
Question If the length of a rectangular parking lot is 10 meters less than twice the width, and the perimeter is 100 meters, find the length of the parking lot
Solution First, write the word problem as an equation. Let x = width of the rectangle Then 2x - 10 = length of the rectangle Then its perimeter is 2x + 2(2x - 10) = 100
The perimeter of a rectangle is computed as 2L + 2W
Simplify the above formula 2x + 4x - 20 = 100 ===> 6x = 120 Therefore x = 20
The length of the rectangle is 30 AND its width is 20.
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Question The perimeter of the quadrilateral is 120cm. Find the lengths and sides. (x+5)(x^2-3)(3x-8)(x+3).
Solution Let us first define how to measure the perimeter of a quadrilateral. The perimeter of any quadrilateral, or any shape with four sides for that matter, is measured by adding the lengths of its sides. Hence, the quadrilateral on the right has a perimeter of 30 which is computed by adding the measure of its sides - 6 + 9 + 7 + 8.
Now, let us go back to our problem. The four sides of the quadrilateral are (x+5), (x^2-3), (3x-8) and (x+3). Hence, its perimeter is computed by (x+5) + (x^2-3) + (3x-8) + (x+3).
To solve for the length of the sides of the quadrilateral:
(x+5) + (x^2-3) + (3x-8) + (x+3) = 120 x^2 - 3+ x+5 + 3x - 8 + x + 3 = 120 x^2 + 5x - 3 = 120 ===> add like terms x^2 + 5x - 123 = 0 ===> right side of the equation is set to zero (x + ?) (x - ?) = 0 ====> to arrive at the root of the quadratic equation, think of two numbers that when you multiply them yields -123 and when you add them gives 5. Hmmm...difficult, however nothing prevents us to looking at a solution which is NOT a whole number.
Now, I got 13.87 and -8.87. Their multiple is -123.02 [close enough] and their sum is 5.
(x + 13.87) (x - 8.87) = 0 x = -13.87 ===> this is NOT an option because it is NEGATIVE. x = 8.87
Now to calculate the lengths of the sides of the quadrilateral: (x+5) ===> 13.87 (x^2-3) ==> 75.68 (3x-8) ===> 18.61 (x+3) ===> 11.87
Now to check whether the sum of the above is 120. 13.87 + 75.68 + 18.61 + 11.87 = 120.02 [close enough]
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