A picture that measures 10 cm by 5 cm is framed with a uniform border that is the same on all sides of the picture. The area of the border is twice the area of the picture Determine the width of the border.
let the width of the border be x m
so the framed picture is 10+2x by 5 + 2x
area of the whole thing = (10+2x)(5 + 2x)
area of picture = 50 cm^2
area of border = (10+2x)(5 + 2x) - 50
It said
(10+2x)(5 + 2x) - 50 = 2(50)
simplify, arrange as a quadratic and solve for x
Hint: make sure to reject the negative value of x
What is 2(50) in the part of teh question?
why did you minus 50
Isn't the area of the whole thing made up of the area of the
picture + the area of the border ?
area of border + 50 = (10+2x)(5 + 2x)
subtract 50 from both sides
area of border = (10+2x)(5 + 2x) - 50
"twice the area of the picture" ----> 2 times 50 = 2(50) = 100
13
To determine the width of the border, we need to set up an equation based on the given information.
Let's assume the width of the border is 'x'.
The dimensions of the frame including the border will then be (10 cm + 2x) by (5 cm + 2x).
Since the area of the border is twice the area of the picture, we can write the equation:
Area of border = 2 * Area of picture
The area of the border is given by:
(10 cm + 2x)(5 cm + 2x) - (10 cm)(5 cm)
Simplifying this equation, we have:
(10 cm + 2x)(5 cm + 2x) - 50 cm² = 2(10 cm)(5 cm)
Expanding the equation:
(50 cm² + 20 cmx + 4x²) - 50 cm² = 2(10 cm)(5 cm)
Simplifying further:
20 cmx + 4x² = 100 cm²
Dividing both sides of the equation by 4:
5 cmx + x² = 25 cm²
Rearranging the equation to get it in quadratic form:
x² + 5 cmx - 25 cm² = 0
Now, we can solve this equation using factoring or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b² - 4ac))/(2a)
In this case, a = 1, b = 5 cm, and c = -25 cm².
Substituting the values in the formula:
x = (-(5 cm) ± √((5 cm)² - 4(1)(-25 cm²)))/(2(1))
Simplifying further:
x = (-5 cm ± √(25 cm² + 100 cm²))/(2)
x = (-5 cm ± √(125 cm²))/(2)
Since we're looking for a positive value for x, we can ignore the negative solution:
x = (-5 cm + √(125 cm²))/(2)
Calculating the square root:
x = (-5 cm + √(125) cm)/(2)
x = (-5 cm + 11.18 cm)/(2)
x ≈ 3.59 cm
Therefore, the width of the border is approximately 3.59 cm.