calculate the slant height for the given square pyramid.
Round to the nearest tenth.
Pyramid base= 6cm
height=5 cm
6.2 cm
5.8 cm
7.8 cm
7.2 cm
Calculate the length of the diagonal for the given rectangular prism.
Round to the nearest tenth.
Length= 10 cm
widith= 4cm
height= 10cm
14.7 cm
10.8 cm
12.2 cm
15.6 cm
Calculate the length of the diagonal for the given rectangular prism.
Round to the nearest tenth.
Length= 14cm
widith= 3
height= 4 cm
First, we will calculate the slant height for the square pyramid:
To solve this, we can use the Pythagorean Theorem (a^2 + b^2 = c^2).
In this case, a is half the base (3 cm) and b is the height (5 cm).
3^2 + 5^2 = c^2
9 + 25 = c^2
34 = c^2
c = √34
c ≈ 5.8 cm
So, the slant height of the pyramid is approximately 5.8 cm.
Next, we will calculate the length of the diagonal for the first rectangular prism:
To solve this, we can use the 3D Pythagorean Theorem (a^2 + b^2 + c^2 = d^2).
In this case, a is the length (10 cm), b is the width (4 cm), and c is the height (10 cm).
10^2 + 4^2 + 10^2 = d^2
100 + 16 + 100 = d^2
216 = d^2
d = √216
d ≈ 14.7 cm
So, the length of the diagonal for the first rectangular prism is approximately 14.7 cm.
Finally, we will calculate the length of the diagonal for the second rectangular prism:
To solve this, we can use the 3D Pythagorean Theorem (a^2 + b^2 + c^2 = d^2) again.
In this case, a is the length (14 cm), b is the width (3 cm), and c is the height (4 cm).
14^2 + 3^2 + 4^2 = d^2
196 + 9 + 16 = d^2
221 = d^2
d = √221
d ≈ 14.9 cm
So, the length of the diagonal for the second rectangular prism is approximately 14.9 cm.
To calculate the slant height of a square pyramid, we can use the Pythagorean theorem.
Given:
Pyramid base = 6 cm
Height = 5 cm
To find the slant height, we need to first find the length of the slant, which is the hypotenuse of the right triangle formed by the height and the half of the pyramid base.
Step 1: Find the length of half of the pyramid base
Half of the pyramid base = 6 cm / 2 = 3 cm
Step 2: Use the Pythagorean theorem to find the length of the slant
Slant Length^2 = Height^2 + (Half of the pyramid base)^2
Slant Length^2 = 5 cm^2 + 3 cm^2
Slant Length^2 = 25 cm^2 + 9 cm^2
Slant Length^2 = 34 cm^2
Step 3: Take the square root of both sides to find the slant length
Slant Length ≈ √34 cm
Slant Length ≈ 5.8 cm
Therefore, the slant height of the given square pyramid is approximately 5.8 cm.
Now, moving on to the second problem:
Given:
Length = 10 cm
Width = 4 cm
Height = 10 cm
To find the length of the diagonal of a rectangular prism, we can use the Pythagorean theorem again.
Step 1: Find the length of the diagonal of the rectangular base
Diagonal of the base^2 = Length^2 + Width^2
Diagonal of the base^2 = 10 cm^2 + 4 cm^2
Diagonal of the base^2 = 100 cm^2 + 16 cm^2
Diagonal of the base^2 = 116 cm^2
Step 2: Find the length of the diagonal of the rectangular prism
Diagonal^2 = Diagonal of the base^2 + Height^2
Diagonal^2 = 116 cm^2 + 10 cm^2
Diagonal^2 = 126 cm^2
Step 3: Take the square root of both sides to find the diagonal length
Diagonal ≈ √126 cm
Diagonal ≈ 11.2 cm
Therefore, the length of the diagonal for the given rectangular prism is approximately 11.2 cm.
Moving on to the third problem:
Given:
Length = 14 cm
Width = 3 cm
Height = 4 cm
We will follow the same steps as before.
Step 1: Find the length of the diagonal of the rectangular base
Diagonal of the base^2 = Length^2 + Width^2
Diagonal of the base^2 = 14 cm^2 + 3 cm^2
Diagonal of the base^2 = 196 cm^2 + 9 cm^2
Diagonal of the base^2 = 205 cm^2
Step 2: Find the length of the diagonal of the rectangular prism
Diagonal^2 = Diagonal of the base^2 + Height^2
Diagonal^2 = 205 cm^2 + 4 cm^2
Diagonal^2 = 209 cm^2
Step 3: Take the square root of both sides to find the diagonal length
Diagonal ≈ √209 cm
Diagonal ≈ 14.7 cm
Therefore, the length of the diagonal for the given rectangular prism is approximately 14.7 cm.
To calculate the slant height of a square pyramid, you can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the height and half the base length of the pyramid.
In this case, the base length is given as 6 cm, and the height is given as 5 cm.
To find the slant height:
1. Divide the base length by 2: 6 cm / 2 = 3 cm
2. Square the base length divided by 2: (3 cm)^2 = 9 cm^2
3. Square the height: (5 cm)^2 = 25 cm^2
4. Add the squares from step 2 and step 3: 9 cm^2 + 25 cm^2 = 34 cm^2
5. Take the square root of the sum from step 4: √(34 cm^2) = 5.8 cm
Therefore, the slant height of the square pyramid is approximately 5.8 cm, which corresponds to the option: 5.8 cm.
In the second question, you are provided with the dimensions of a rectangular prism (length, width, and height) and asked to calculate the length of the diagonal.
To find the length of the diagonal:
1. Square the length: (10 cm)^2 = 100 cm^2
2. Square the width: (4 cm)^2 = 16 cm^2
3. Square the height: (10 cm)^2 = 100 cm^2
4. Add the squares from steps 1, 2, and 3: 100 cm^2 + 16 cm^2 + 100 cm^2 = 216 cm^2
5. Take the square root of the sum from step 4: √(216 cm^2) ≈ 14.7 cm
Therefore, the length of the diagonal of the rectangular prism is approximately 14.7 cm, which corresponds to the option: 14.7 cm.
In the third question, you are provided with the dimensions of another rectangular prism (length, width, and height) and asked to calculate the length of the diagonal.
To find the length of the diagonal:
1. Square the length: (14 cm)^2 = 196 cm^2
2. Square the width: (3 cm)^2 = 9 cm^2
3. Square the height: (4 cm)^2 =16 cm^2
4. Add the squares from steps 1, 2, and 3: 196 cm^2 + 9 cm^2 + 16 cm^2 = 221 cm^2
5. Take the square root of the sum from step 4: √(221 cm^2) ≈ 14.9 cm
Therefore, the length of the diagonal of the rectangular prism is approximately 14.9 cm, which is not included in the given options.