Geometry help please!?

Question

1.A pyramid has a volume of 76in^3 find the volume of this pyramid if its sides are dilated by a scale factor of 2

2. Find all possible integer lengths of the third side of a triangle if the first two sides have lengths of 7 and 5

Answer 1

 Given that volume of a pyramid = 76in^3

Given that its scale factors are dialted by 2

Volume of a pyramid = 1/3(base area * hieght)

                                  = 1/3( (1/2 b*h) *h)  ( pyramid will be in triangle shape)

                                 = 1/6 (b * h * h)

Given that sides are dilated by 2

1/6 (b/2 * h/2 * h/2)

We have volume of a pyramid = 76in^3

That means value of (1/6 * b * h * h) = 76in^3

                                                           = 76/8 = 9.5cub.inches

Therefore the volume of the given pyramid if its sides are dilated by a scale factor of 2 = 9.5cub.in

 

 

 

Comments

(1). Let assume that pyramid is rectangular-based pyramid.

Therefore the volume of the given pyramid if its sides are dilated by a scale factor of 2 = 19 cub.in.

In case pyramid is triangular pyramid pyramid.

The formula for the volume of rectangular-based pyramid V = 1/3 (A * H), where A is base area and H is the height of the pyramid.

Here base is triangle, so the area of triangle A = 1/2 (b * h), where b is the base and h is height of the triangle.

Therefore, volume of the rectangular-based pyramid V = 1/3 (A * H) = 1/3[ (1/2 * b * h) * H ] = 1/6 (b * h * H).

We have volume of a pyramid = 76 in³.

That means value of V = 1/6 * b * h * H = 76in^3

Given that sides are dilated by 2.

V = 1/6 * b/2 * h * H

V = 1/2 (1/6 * b * h * H )

V = 1/2 (76)

V = 38.

Therefore the volume of the given pyramid if its sides are dilated by a scale factor of 2 = 38 cub.in.

(2).The possible integer values engths of the third side of a triangle are s = 3, 4, 5, 6, 7, 8, 9, 10, 11.

Answer 2

(1).

The volume of the pyramid is 76 in³.

Let assume that pyramid is rectangular-based pyramid.

The formula for the volume of rectangular-based pyramid V = 1/3 (A * H), where A is base area and H is the height of the pyramid.

Here base is rectangle, so the area of rectangle A = l * w, where l is the length and w is width.

Therefore, volume of the rectangular-based pyramid V = 1/3 (A * H) = 1/3[ (l * w) * H ] = 1/3 (l * w * H).

We have volume of a pyramid = 76 in³.

That means value of V = 1/3 * l * w * H = 76in^3

Given that sides are dilated by 2.

V = 1/3 ( l/2 * w/2 * H )

V = 1/4 (76)

V = 19.

Therefore the volume of the given pyramid if its sides are dilated by a scale factor of 2 = 19 cub.in.

(2).

Triangle inequality property : Any side of a triangle can’t be more than the sum of its other two sides or less than there difference.

For any three sides of a triangle, the following holds: a + b > c.

The two sides of the triangle are 7 and 5.

Let s be the third side of the triangle.

Now if we let a, b and c be each possible permutation of 7, 5 and s, you get:

(1) 7 + s > 5

(2) 5 + s > 7

(3) 7 + 5 > s

We can reduce them to this:

(1) s > - 2

(2) s > 2

(3) s < 12

This gives the possible range 2 < s < 12.

The possible integer values are s = 3, 4, 5, 6, 7, 8, 9, 10, 11.