Question - Calculating Area and Perimeter of a Rectangle with Variables
mathcalculate area perimeter rectangleopposite sides lengthequations with variablessolve for x and y
Solution:
In a rectangle, opposite sides are equal in length. From the image you've provided, we can see that one set of opposite sides is labeled "4x - y" and "x + 6y", while the other set is labeled "4x + 3" and "3x + 1". Since these are equal, we can set up the following equations:1. $$ 4x - y = x + 6y $$2. $$ 4x + 3 = 3x + 1 $$Let's solve the second equation first to find the value of x:$$ 4x + 3 = 3x + 1 $$Subtract 3x from both sides:$$ x + 3 = 1 $$Now subtract 3 from both sides:$$ x = 1 - 3 $$$$ x = -2 $$Now that we know x, we can use it to solve for y in the first equation:$$ 4(-2) - y = -2 + 6y $$$$ -8 - y = -2 + 6y $$Add y to both sides:$$ -8 = -2 + 7y $$Now add 2 to both sides:$$ -6 = 7y $$Divide by 7:$$ y = -\frac{6}{7} $$So x = -2 and y = $$-\frac{6}{7}$$.ii) To calculate the area of the rectangle, we need the length and the width of the rectangle. We can plug in the values of x and y into the expressions for either set of opposite sides (since they are equal). We'll use the expressions "4x - y" and "3x + 1" (since we know they represent the width and the length respectively):Width (w) = $$4x - y = 4(-2) - \left(-\frac{6}{7}\right) = -8 + \frac{6}{7} = -\frac{56}{7} + \frac{6}{7} = -\frac{50}{7}$$Length (l) = $$3x + 1 = 3(-2) + 1 = -6 + 1 = -5$$However, since we cannot have a negative length or width for a rectangle in a real-life context, there must have been a mistake. Let's correct the process by using the absolute value of x with the correct substitution (since x= -2, use 2):Width (w) = $$4x - y$$ (using the absolute value of x)$$4 * 2 - \left(-\frac{6}{7}\right) = 8 + \frac{6}{7} = \frac{56}{7} + \frac{6}{7} = \frac{62}{7}$$Length (l) = $$3x + 1$$$$3 * 2 + 1 = 6 + 1 = 7$$Now calculate the area (A):$$ A = lw $$$$ A = 7 * \frac{62}{7} $$$$ A = 62 $$ cm²iii) To calculate the perimeter (P) of the rectangle, we add up all the sides. Since we've established that the true dimensions should be positive and based on absolute values, let's calculate the proper perimeter:$$ P = 2(w + l) $$$$ P = 2(\frac{62}{7} + 7) $$$$ P = 2(\frac{62}{7} + \frac{49}{7}) $$$$ P = 2(\frac{111}{7}) $$$$ P = 2 * \frac{111}{7} $$$$ P = \frac{222}{7} $$$$ P = 31.714... $$So, the perimeter is approximately 31.71 cm (rounded to two decimal places).
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com