Question

Work out the volume of the cone, giving your answer

to 3 significant figures.
24 cm
7 cm
25.cm

Answer 1

If the given lengths of 7 cm, 24 cm, and 25 cm for the sides of a triangle satisfy the equation of the Pythagorean Theorem, then, yes, the given side lengths are those of a right triangle. Let's see if they are:

The well-known equation of the famous Pythagorean Theorem is:

a² + b² = c², which says that for a right triangle, the sum of the squares of the lengths of the two shorter sides of the triangle is equal to the square of the length of the longest side called the hypotenuse, where a and b are the lengths of the two shorter sides (also called the "legs") and c is the length of the hypotenuse (the side opposite the right angle).

We're given that a = 7 cm and b = 24 cm and that c = 25 cm. Substituting these values into the equation of the Pythagorean Theorem, we get:

a² + b² = c²

(7 cm)² + (24 cm)² = (25 cm)²

(7 cm)(7 cm) + (24 cm)(24 cm) = (25 cm)(25 cm)

49 cm² + 576 cm² = 625 cm²

625 cm² = 625 cm²

As we can see, the equation of the Pythagorean Theorem is satisfied, i.e., made a true statement, by the given lengths; therefore, if these three lengths, 7 cm, 24 cm, and 25 cm, are the lengths of the sides of a triangle, then the triangle is indeed a right triangle.

Answer 2

The volume of the cone is approximately
`\(4080 \, \text{cm}^3\)` rounded to three significant figures. The cone has a radius of 7 cm and a height of 25 cm.

Explanation:

To calculate the volume of a cone, you can use the formula:

`\[ V = (1)/(3) \pi r^2 h \]`

where:

( V ) is the volume,

( pi ) is a mathematical constant (approximately 3.14159),

( r ) is the radius of the base of the cone,

( h ) is the height of the cone.

In your case, the given values are:

`\( r = 7 \)` cm (radius),

`\( h = 25 \)` cm (height).

Now, plug these values into the formula:

`\[ V = (1)/(3) \pi (7^2) * 25 \]`

Calculate the result:

`\[ V = (1)/(3) \pi * 49 * 25 \]`

`\[ V = (1)/(3) * 1225 \pi \]`

`\[ V \approx 4083.67 \, \text{cm}^3 \]`

Rounded to three significant figures, the volume of the cone is approximately
`\(4080 \, \text{cm}^3\)`.