Question

Given the volume of the cuboid is atleast 150cm3 and the surface area of the cuboid is less than 320cm2. Determine all the possible values of x by solving simultaneous inequalities.

Answer 1

The volume of a cuboid is given by the formula V = l * w * h, where V is the volume, l is the length, w is the width, and h is the height.

The surface area of a cuboid is given by the formula A = 2lw + 2lh + 2wh, where A is the surface area and l, w, and h are the dimensions of the cuboid as described above.

We can use these formulas to write two inequalities:

V = l * w * h >= 150
A = 2lw + 2lh + 2wh < 320

To solve this system of inequalities, we can begin by substituting the expression for V into the second inequality:

2lw + 2lh + 2wh < 320
l * w * h >= 150

Next, we can divide both sides of the second inequality by l * w * h to find an expression for h in terms of l and w:

h < 320 / (l * w)
h >= 150 / (l * w)

Since h must be greater than or equal to 150 / (l * w) and less than 320 / (l * w), we can find all possible values of h by finding the values of l and w that satisfy these inequalities.

To find the values of l and w that satisfy these inequalities, we can begin by substituting the expression for h into the first inequality:

2lw + 2l(150 / (l * w)) + 2w(150 / (l * w)) < 320
3lw + 300 < 320
lw < 20

Since lw must be less than 20, we can find all possible values of l and w by listing all pairs of values that satisfy this inequality. Some possible pairs of values are (1, 20), (2, 10), (4, 5), and so on.

We can then use these values of l and w to find the corresponding values of h using the expression h = 150 / (l * w). For example, if l = 1 and w = 20, then h = 150 / (1 * 20) = 7.5. If l = 2 and w = 10, then h = 150 / (2 * 10) = 15. If l = 4 and w = 5, then h = 150 / (4 * 5) = 7.5.

Therefore, all possible values of x (which represents the length, width, and height of the cuboid) are:

(1, 20, 7.5)
(2, 10, 15)
(4, 5, 7.5)
and so on…

Answer 2

Explanation:

To solve this problem, we can start by finding the possible range of values for the volume of the cuboid, which is at least 150 cm^3. The volume of a cuboid is given by the formula V = lwh, where l is the length, w is the width, and h is the height. Since the volume is at least 150 cm^3, we can set up the following inequality:

V >= 150

lwh >= 150

Next, we can find the possible range of values for the surface area of the cuboid, which is less than 320 cm^2. The surface area of a cuboid is given by the formula A = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height. Since the surface area is less than 320 cm^2, we can set up the following inequality:

A < 320

2lw + 2lh + 2wh < 320

Now, we can solve these simultaneous inequalities to find the possible values of x, which represent the lengths, widths, and heights of the cuboid. To do this, we can start by finding the possible values of l, w, and h that satisfy both inequalities.

One way to do this is to rewrite the inequalities in terms of a single variable, such as l, and then solve for the possible values of l that satisfy both inequalities. For example, we could substitute the expression for the volume (lwh) and the expression for the surface area (2lw + 2lh + 2wh) into the inequality for the volume (V >= 150) and the inequality for the surface area (A < 320), respectively, and then solve for the possible values of l that satisfy both inequalities.

I hope this helps! Let me know if you have any questions or need further assistance.