The area of a rectangle is 105mm^2 . The length is six millimeters less than three times its width. Find the dimensions of the rectangle.
a) Write the length in terms of its width(use w as the variable).
b) Write an equation using w for width.
equation:
c) What should the dimensions be?
l=
w=
a) The length is six millimeters less than three times its width. Let the width be w. The length would then be 3w - 6.
b) The equation for the area of a rectangle is length times width, so the equation would be (3w - 6)w = 105.
c) To solve the equation, we can first multiply out the brackets: 3w^2 - 6w = 105. Rearranging the equation we get: 3w^2 - 6w - 105 = 0.
Next, we can try to factorize the quadratic equation. In this case, it doesn't factorize neatly, so we can use the quadratic formula: w = (-b ± sqrt(b^2 - 4ac)) / 2a. Plugging in the values from our equation, we get:
w = (-(-6) ± sqrt((-6)^2 - 4(3)(-105))) / 2(3)
w = (6 ± sqrt(36 + 1260)) / 6
w = (6 ± sqrt(1296)) / 6
w = (6 ± 36) / 6
This gives us two possible values for w: w = 7 or w = -5.
Since the width of a rectangle cannot be negative, we can disregard the negative value. Therefore, the width of the rectangle is 7.
To find the length, we can substitute this value back into our equation for the length: l = 3w - 6 = 3(7) - 6 = 21 - 6 = 15.
Therefore, the dimensions of the rectangle are:
l = 15 mm
w = 7 mm