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

A.) To write the area as the product of two binomials with integer coefficients, we need to factorize the given expression:

12a^2 - 5a - 2m^2

The factors will have the form (ma + n)(oa + p), where m, n, o, and p are integers. We need to find the values of m, n, o, and p.

By analyzing the expression, we can see that the coefficient of the squared term (a^2) is 12, the coefficient of the linear term (a) is -5, and there is no constant term.

To factorize, we need to find two numbers whose product is equal to the product of the squared term (12a^2) and whose sum is equal to the coefficient of the linear term (-5a).

By trial and error, we find that 4 and 3 satisfy these conditions since 4 * 3 = 12 and 4 + 3 = 7.

Therefore, the area can be factored as:

(4a - 1)(3a + 2)

B.) To find the area of the path, we need to add the width of the path to the length and width of the garden.

Let's assume the length of the garden is L and the width is W.

The new length including the path will be L + 2 (since there is a 1m width path on both sides of the garden).

The new width including the path will be W + 2.

The area of the path will be the difference between the area of the outer rectangle and the area of the inner rectangle (the garden).

Area of the path = (L + 2)(W + 2) - LW
= LW + 2L + 2W + 4 - LW
= 2L + 2W + 4

Simplifying the expression, we get:
Area of the path = 2(L + W + 2)

C.) To calculate the volume of concrete used, we need to multiply the area of the path by the depth of the concrete (10 cm) and convert it to cubic meters.

Given that a = 6, we substitute it into the expression for the area of the path we got earlier:

Area of the path = 2(L + W + 2)
= 2(6 + W + 2)
= 2(8 + W)
= 16 + 2W

The volume of concrete used is then:

Volume = (16 + 2W) * 10 cm^3

Since 1 meter is equal to 100 cm, we can convert the depth of concrete to meters:

Volume = (16 + 2W) * 0.1 m^3
= 1.6 + 0.2W m^3

A.) To write the area as the product of two binomials with integer coefficients, we need to factor the given expression.

12a^2 - 5a - 2m^2 can be factored as (4a + 1m)(3a - 2m).

So the area of the rectangular garden can be written as the product of two binomials with integer coefficients: (4a + 1m)(3a - 2m).

B.) To find the area of the path, we need to add the width of the path to the length and width of the garden.

Let's assume the length of the garden is L and the width is W. The length including the path would then be L + 2 and the width including the path would be W + 2.

The area of the garden is L * W, and the area of the garden including the path is (L + 2)(W + 2).

To find the area of the path, we subtract the area of the garden from the area of the garden including the path:

Area of the path = (L + 2)(W + 2) - L * W

Simplifying this expression gives us the final answer for the area of the path.

C.) To calculate the volume of concrete used, we need to find the volume of the path.

The volume of a rectangular prism is given by multiplying the length, width, and height. In this case, the length and width of the path are L + 2 and W + 2, and the height is the depth of the concrete poured.

The volume of the path can be calculated as (L + 2)(W + 2)(0.1) since the depth is given in centimeters and we need to convert it to meters (1 m = 100 cm).

Given that a = 6, you will substitute this value into the expressions for L and W in the area of the path and the volume of concrete used to get the final numerical values.

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