A rectangular garden has an area if
12a^2-5a-2m^2
A.) Write the area as the product of two binomials with integer coefficients
B.) the garden is to be completely enclosed by a path 1 m wide. find and simplify an expression for the area of the path
C.) the path is concrete, poured to a depth of 10 cm. calculate the volume in m^3 of concrete used if a=6
I think you meant
12a^2 - 5a - 2 = (3a-2)(4a+1)
After the path, each dimension is increased by 2, so the new total area is
(3a-2+2)(4a+1+2) = (3a)(4a+3)
Now subtract the original area to get the area of the path:
(3a)(4a+3)-(3a-2)(4a+1)
now just simplify that
Finally, just plug in a, recalling that the volume
v = (a)(.10) m^3
A.) To write the area as the product of two binomials with integer coefficients, we can factorize the expression "12a^2 - 5a - 2m^2". After some calculations, it can be written as (4a - 1)(3a + 2m).
B.) To find the area of the path, we need to add the width of the path on both sides of the rectangular garden. Since the width of the path is 1m, the length of the new rectangle will increase by 2m, and the width will increase by 2m as well. Therefore, the area of the path can be calculated as (length + 2)(width + 2) - length * width.
C.) To calculate the volume of concrete used, we need to find the volume of the path. The volume of the path can be calculated by multiplying the area of the path by the depth of the concrete. Since the path is poured to a depth of 10cm, which is equivalent to 0.1m, the volume can be calculated as: (area of the path) * (0.1m).
A.) To write the area as the product of two binomials with integer coefficients, we need to factor the expression 12a^2 - 5a - 2m^2.
The factored form of the expression is:
(3a - 2m)(4a + m)
B.) The area of the path can be calculated by subtracting the area of the garden from the area of the larger rectangle formed by adding 2 meters to each side of the garden.
Let's first find the area of the larger rectangle:
Area of larger rectangle = (length + 2)(width + 2)
Since we don't have the dimensions of the rectangle, we'll use variables to represent them:
Length of the garden = a
Width of the garden = b (unknown)
Area of larger rectangle = (a + 2)(b + 2)
Now, let's find the area of the path by subtracting the area of the garden from the area of the larger rectangle:
Area of path = (a + 2)(b + 2) - 12a^2 + 5a + 2m^2
C.) The volume of concrete used can be calculated by multiplying the area of the path by the depth.
We know that the depth is 10 cm, which is equivalent to 0.1 m.
Volume of concrete used = Area of path * depth
Volume of concrete used = (a + 2)(b + 2) - 12a^2 + 5a + 2m^2 * 0.1
If a = 6, you can substitute it into the expressions above to find the specific values.
A.) To write the area as the product of two binomials with integer coefficients, we need to express the given expression in factored form. To do this, we can use the technique called factorization.
The given expression is: 12a^2 - 5a - 2m^2
To factorize this expression, we need to find two binomials whose product gives us the given expression. We'll start by multiplying the coefficient of the first term (12), and the constant term (-2m^2). The result is -24m^2.
Next, we find two numbers whose sum is the coefficient of the middle term (-5a) and whose product is the above result (-24m^2). After some calculation, we find that -8m^2 and +3m^2 satisfy these conditions.
So now, we can rewrite the area as a product of two binomials:
(4a + m)(3a - 2m)
B.) To find and simplify an expression for the area of the path, we need to add the width of the path (which is 1m) to both the length and width of the rectangular garden.
If the length of the garden is L and the width is W, then the length of the garden plus the path will be (L + 2) and the width of the garden plus the path will be (W + 2).
Therefore, the expression for the area of the path is:
(L + 2)(W + 2)
C.) To calculate the volume of concrete used if a = 6, we need to consider that the depth of the concrete poured is 10 cm, which is equal to 0.1 m.
The volume of concrete used is given by the product of the area of the path (L + 2)(W + 2) and the depth of the concrete (0.1 m).
So, the volume of concrete used if a = 6 is:
(6 + 2)(W + 2)(0.1) = 8(W + 2)(0.1) = 0.8(W + 2) m^3