A rectangle is 4 inches longer than it is wide and has an area of 165 square inches. Find its dimensions
165 = 11*15
Hmm, finding the dimensions of a rectangle can be quite rectangular-ly confusing sometimes, but fear not, I'm here to help! Let's approach this with a touch of mathematical humor, shall we?
Let's say the width of the rectangle is "W" (in inches). Since the rectangle is 4 inches longer than it is wide, we can express its length (L) as "W + 4."
Now, we know that the area of a rectangle is given by the formula: Area = Length × Width. In this case, the area is 165 square inches. So we have the equation:
165 = (W + 4)W
If we expand this equation, we get:
165 = W^2 + 4W
Now, let's rearrange this equation to make it quadratic:
W^2 + 4W - 165 = 0
To solve this equation, we can either factor it, use the quadratic formula, or consult with a mathemagician! Let's go with factoring:
(W + 15)(W - 11) = 0
Now, if either (W + 15) or (W - 11) equals zero, we'll have a solution. So we have two possibilities:
W + 15 = 0 or W - 11 = 0
Solving these equations gives us:
W = -15 or W = 11
Since we're talking about measurements in inches, let's discard the negative solution. So, W = 11.
Remember, the length (L) is equal to the width (W) plus 4. So, L = 11 + 4 = 15.
Therefore, the dimensions of the rectangle are 11 inches by 15 inches.
And that, my friend, is how this rectangle's mysterious dimensions were unraveled. Ta-da!
Let's assume the width of the rectangle is x inches. Since the rectangle is 4 inches longer than it is wide, the length can be expressed as x + 4 inches.
To find the area of a rectangle, we can use the formula A = length * width.
Given that the area of the rectangle is 165 square inches, we can write the equation:
165 = (x + 4) * x
Simplifying:
165 = x^2 + 4x
Rearranging to standard quadratic form:
x^2 + 4x - 165 = 0
This equation can be solved by factoring, completing the square, or using the quadratic formula. Let's use factoring:
Factoring the equation:
(x + 15)(x - 11) = 0
Setting each factor equal to zero and solving for x:
x + 15 = 0 or x - 11 = 0
Solving:
x = -15 or x = 11
Since the width of the rectangle cannot be negative, we ignore the negative solution.
Therefore, the width of the rectangle is 11 inches.
To find the length, we can substitute the value of x back into our expression for length:
Length = x + 4
Length = 11 + 4
Length = 15 inches
So, the dimensions of the rectangle are 11 inches by 15 inches.
To find the dimensions of the rectangle, we can set up a system of equations based on the given information.
Let's assume the width of the rectangle as "w" inches.
According to the given information, the length of the rectangle is 4 inches longer than its width, so the length would be "w + 4" inches.
The formula for the area of a rectangle is length multiplied by width. In this case, the area is given as 165 square inches.
Using this information, we can set up the equation:
Length × Width = Area
(w + 4) × w = 165
Now, let's solve this equation to find the width of the rectangle:
w(w + 4) = 165
w^2 + 4w = 165
w^2 + 4w - 165 = 0
This is a quadratic equation in standard form. To solve it, there are several methods, such as factoring, completing the square, or using the quadratic formula. In this case, we'll use factoring.
We can start by looking for two numbers whose product is -165 and whose sum is 4. After trying different combinations, we find that the numbers 11 and -15 satisfy these conditions.
So, we can rewrite the equation as:
(w + 11)(w - 15) = 0
Now, we can set each factor equal to zero and solve for "w":
w + 11 = 0 or w - 15 = 0
If we solve both equations, we'll find two possible values for the width:
w = -11 or w = 15
Since we're dealing with dimensions, a negative value doesn't make sense. Therefore, the width of the rectangle is 15 inches.
Now, we can find the length by substituting the width into our earlier expression:
Length = w + 4
Length = 15 + 4
Length = 19 inches
So, the dimensions of the rectangle are width = 15 inches and length = 19 inches.