the function for the perimeter of a rectangle is given as P=4x+6
a) Give an expression for the length of the rectangle in terms of the width.
b) If the width is half the length, state a function for both the width and length describing the domain for which these fuctions exist.
a) If P = 4x + 6, since P = 2L + 2W, then the two sides will be W = x and L = x +3.
b) W = 1/2 L = 1/2(x + 3)
I hope this helps.
To find the expressions for the length and width of the rectangle, we need to solve the given equation for one of the variables.
a) Expression for the length in terms of the width:
The perimeter of a rectangle is given by the formula P = 2(length + width). In this case, the formula is given as P = 4x + 6. We can equate these two formulas and solve for the length.
2(length + width) = 4x + 6
Divide both sides by 2:
length + width = 2x + 3
Subtract the width from both sides:
length = 2x + 3 - width
Therefore, the expression for the length of the rectangle in terms of the width is:
length = 2x + 3 - width
b) Function for both the width and length when the width is half the length:
If the width is half the length, we can express this relationship as width = length/2.
We already have an expression for the length in terms of the width:
length = 2x + 3 - width
Substitute width = length/2 into the expression for length:
length = 2x + 3 - length/2
Multiply both sides by 2 to eliminate the fraction:
2 * length = 4x + 6 - length
Combine like terms:
3 * length = 4x + 6
Divide both sides by 3:
length = (4x + 6)/3
The function for the length is length = (4x + 6)/3, and the function for the width is width = length/2.
As for the domain, the functions exist for all values of x since there are no restrictions given in the problem.