Mang Jose wants to make a table which has an area of 6m^2.The length of the table has to be 1m longer than the width.
a.If the width of the table is p meters,what will be it's length.
b.Form a quadratic equation for the problem and give the dimensions of the table.
width ---- x m
length = x+1 m
area = x(x+1)
x(x+1) = 6
expand, arrange in the usual quadratic form, and solve.
I assume you know how to solve basic quadratics, or else you wouldn't have these
types of problems.
a. If the width of the table is p meters, then the length of the table will be p + 1 meters.
b. To form a quadratic equation, we can use the area formula for a rectangle: Area = Length × Width.
Since the area of the table is given as 6m^2 and the length is p + 1, we can set up the equation as:
6 = (p + 1) × p
Expanding the equation:
6 = p^2 + p
Rearranging the equation to standard quadratic form:
p^2 + p - 6 = 0
To solve this quadratic equation, we can factor it:
(p + 3)(p - 2) = 0
Setting each factor equal to zero:
p + 3 = 0 or p - 2 = 0
Solving for p:
p = -3 or p = 2
Since width cannot be negative, we discard the negative value. Therefore, the width of the table is 2 meters.
Using the width, we can calculate the length:
Length = Width + 1 = 2 + 1 = 3 meters.
So, the dimensions of the table are 2 meters width and 3 meters length.
To find the dimensions of the table, we need to solve the problem using algebraic equations.
Let's start with the given information:
1. The area of the table is 6m^2.
2. The length of the table is 1m longer than the width.
a. To find the length of the table (L), given the width (p), we can set up an equation using the area formula for a rectangle: Area = Length × Width.
So, in this case, the equation becomes:
6m^2 = L × p
Now, using the second given information, we know that the length (L) is 1m longer than the width (p).
So, we can express the length as p + 1.
Substituting this into the equation, we have:
6m^2 = (p + 1) × p
b. To form a quadratic equation for the problem, we can rewrite the equation in the previous step and put it in standard quadratic form (ax^2 + bx + c = 0). In this case, we have:
p^2 + p - 6 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula. In this case, let's factorize it:
(p + 3)(p - 2) = 0
Setting each factor equal to zero and solving for p, we find two possible solutions:
p + 3 = 0 => p = -3
p - 2 = 0 => p = 2
However, since we are dealing with dimensions, the width cannot be negative. Therefore, the width (p) of the table is 2 meters.
To find the length (L), we can substitute the value of p into the equation L = p + 1:
L = 2 + 1 = 3 meters
Therefore, the dimensions of the table are: width = 2 meters and length = 3 meters.