show me some example of complex border problem in grade 10 math and its answer for praticing
Sure! Here's an example of a complex border problem in Grade 10 math:
Problem:
A rectangular garden is 12 meters long and 8 meters wide. The owner wants to build a rectangular border along the inside of the garden. The width of the border is the same all around and the area of the border is 40 square meters. What is the width of the border?
Solution:
Step 1: Calculate the area of the inner rectangular garden:
Area of the inner garden = length × width
Area of the inner garden = 12 meters × 8 meters
Area of the inner garden = 96 square meters
Step 2: Let's assume the width of the border is 'x' meters.
So, the dimensions of the outer rectangular garden are:
Length = 12 + 2x (the original length plus border on both sides)
Width = 8 + 2x (the original width plus border on both sides)
Step 3: Calculate the area of the outer rectangular garden:
Area of the outer garden = length × width
Area of the outer garden = (12 + 2x) meters × (8 + 2x) meters
Area of the outer garden = 96 + 24x + 16x + 4x^2
Area of the outer garden = 4x^2 + 40x + 96
Step 4: Since the area of the outer garden is the area of the inner garden plus the area of the border (40 square meters), we can set up the equation:
4x^2 + 40x + 96 = 96 + 40
4x^2 + 40x = 40
x^2 + 10x - 10 = 0
Step 5: Use the quadratic formula to solve for 'x':
x = (-b ± √(b^2 - 4ac))/2a
Plugging in the values, we get:
x = (-10 ± √(10^2 - 4(1)(-10)))/(2(1))
x = (-10 ± √(100 + 40))/(2)
x = (-10 ± √140)/2
x = (-10 ± 2√35)/2
x = -5 ± √35
Since the width cannot be negative, the width of the border is √35 - 5 or approximately 0.87 meters.
Therefore, the width of the border is approximately 0.87 meters.