A solid rectangular block has a base that measures 3x cm by 2x cm. The height of the block is ycm and its volume is 72cm3 3 . i. Express y in terms of x. ii...

Answer Details

The volume of a solid rectangular block is given by the formula V = lwh, where l, w, and h are the length, width, and height of the block, respectively. In this problem, we are given that the base of the block has dimensions 3x cm by 2x cm, so we have l = 3x cm and w = 2x cm. The height of the block is y cm, so h = y cm. We are also given that the volume of the block is 72 cm³, so we have:

V = lwh

72 = (3x)(2x)(y)

72 = 6x^2y

Solving for y, we get:

y = 72/6x^2

y = 12/x^2

Therefore, the height of the block is 12/x^2 cm.

b.

To find the total surface area of the solid rectangular block, we need to consider the six faces of the block: the top face, bottom face, front face, back face, left face, and right face.

Given:
Base length = 3x cm
Base width = 2x cm
Height = y cm
Volume = 72 cm^3

The volume of a rectangular block is given by the formula:

Volume = Base Area * Height

Therefore, we can write the equation:

72 cm^3 = (3x cm * 2x cm) * y cm

Simplifying this equation, we have:

72 = 6x^2 * y

Now, let's express the total surface area of the block in terms of x only.

The total surface area of the block can be calculated by adding the areas of all six faces:

Total Surface Area = 2 * (Base Area) + (Front Face Area) + (Back Face Area) + (Left Face Area) + (Right Face Area)

The base area is given by:

Base Area = Length * Width = (3x cm) * (2x cm) = 6x^2 cm^2

The front face and back face both have the same dimensions, so their areas are equal:

Front Face Area = Back Face Area = Length * Height = (3x cm) * (y cm) = 3xy cm^2

Similarly, the left face and right face both have the same dimensions, so their areas are equal:

Left Face Area = Right Face Area = Width * Height = (2x cm) * (y cm) = 2xy cm^2

Now, let's substitute these values into the equation for the total surface area:

Total Surface Area = 2 * (6x^2 cm^2) + 2 * (3xy cm^2) + 2 * (2xy cm^2)

Simplifying further, we have:

Total Surface Area = 12x^2 cm^2 + 6xy cm^2 + 4xy cm^2

Finally, we can express the total surface area of the block in terms of x only as:

Total Surface Area = 12x^2 cm^2 + 10xy cm^2