Square pyramid with base 9 ft and slant height 12.8 ft.

Let’s solve each problem one by one. These are all volume problems for pyramids and cones.

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Problem 5: Square Pyramid

We’re given:
- Base is a square with side = 9 ft
- Slant height (the diagonal edge from base to apex) = 12.8 ft — but wait! For volume, we need the vertical height, not slant height.

Looking at the diagram, there’s a right triangle inside: half the base (4.5 ft), vertical height (h), and slant height (12.8 ft).

Use Pythagorean theorem:

> h² + 4.5² = 12.8²
> h² + 20.25 = 163.84
> h² = 163.84 - 20.25 = 143.59
> h ≈ √143.59 ≈ 11.98 ft → let’s use 12 ft for simplicity? Wait — better to calculate precisely.

Actually, let’s compute exactly:

√143.59 ≈ 11.983 → so about 11.98 ft

Volume of pyramid = (1/3) × base area × height
Base area = 9 × 9 = 81 ft²
Height ≈ 11.98 ft

Volume ≈ (1/3) × 81 × 11.98 = 27 × 11.98 = let’s compute:

27 × 12 = 324 → minus 27 × 0.02 = 0.54 → so 324 - 0.54 = 323.46? No — wait, 11.98 is less than 12, so:

Better: 27 × 11.98 = 27 × (12 - 0.02) = 324 - 0.54 = 323.46 ft³

But maybe the diagram intends us to use the labeled “height” as 12.8? Wait — no, in the diagram, 12.8 is labeled on the slanted edge, and there’s a dashed line showing the vertical height with a right angle mark — so yes, we must calculate it.

Wait — actually, looking again: the label “12.8 ft” is pointing to the slant height (edge of triangular face), and the vertical height is unlabeled — so we did correct.

But perhaps the problem expects us to use 12.8 as the height? Let me check the diagram description again.

In the user’s image description: “12.8 ft” is written along the slanted edge — that’s slant height. The vertical height is shown with a perpendicular symbol — so we must calculate it.

So:

h = √(12.8² - 4.5²) = √(163.84 - 20.25) = √143.59 ≈ 11.983 ft

Volume = (1/3) * 81 * 11.983 ≈ 27 * 11.983 = let's do exact multiplication:

11.983 × 27:

10 × 27 = 270
1.983 × 27 = (2 × 27) - (0.017 × 27) = 54 - 0.459 = 53.541
Total = 270 + 53.541 = 323.541 ft³

Rounded to nearest tenth: 323.5 ft³

But maybe they expect exact value? Or perhaps I misread.

Wait — another thought: sometimes in diagrams, if they label the height directly, but here it’s clearly on the slant. Let me assume our calculation is correct.

Alternatively, perhaps the 12.8 is the height? But the arrow points to the edge — so likely slant.

I’ll go with calculated height.

But let’s double-check with calculator-style:

12.8 squared = 163.84
4.5 squared = 20.25
Difference = 143.59
Square root = approx 11.9833

Then volume = (1/3)*81*11.9833 = 27 * 11.9833 = 323.5491 → so 323.5 ft³

Okay.

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Problem 6: Cone

Given:
- Diameter = 17 km → radius r = 8.5 km
- Height h = 23.6 km

Volume of cone = (1/3)πr²h

First, r² = 8.5² = 72.25

Then, r²h = 72.25 × 23.6

Calculate that:

72.25 × 20 = 1445
72.25 × 3.6 = let’s see: 72.25 × 3 = 216.75; 72.25 × 0.6 = 43.35 → total 216.75 + 43.35 = 260.1
So total r²h = 1445 + 260.1 = 1705.1

Now multiply by π: 1705.1 × π ≈ 1705.1 × 3.1416 ≈ ?

First, 1700 × 3.1416 = 5340.72
5.1 × 3.1416 ≈ 16.022
Total ≈ 5356.742

Now divide by 3: 5356.742 / 3 ≈ 1785.58 km³

More accurately:

Use direct formula:

V = (1/3) * π * (8.5)^2 * 23.6

Compute step by step:

8.5^2 = 72.25

72.25 * 23.6 = let's compute exactly:

72.25 * 20 = 1445
72.25 * 3 = 216.75
72.25 * 0.6 = 43.35
Sum: 1445 + 216.75 = 1661.75 + 43.35 = 1705.1

Now 1705.1 * π = 1705.1 * 3.1415926535 ≈

Let me compute:

1705.1 * 3 = 5115.3
1705.1 * 0.1415926535 ≈ ?

First, 1705.1 * 0.14 = 238.714
1705.1 * 0.0015926535 ≈ 1705.1 * 0.0016 = 2.72816, minus a bit → say 2.715

So total ≈ 238.714 + 2.715 = 241.429

Thus total π part ≈ 5115.3 + 241.429 = 5356.729

Divide by 3: 5356.729 / 3 = 1785.5763...

So approximately 1785.6 km³

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Problem 7: Hexagonal Pyramid

Given:
- Base area B = 247.75 in²
- Height = 19.5 in

Volume = (1/3) * B * h = (1/3) * 247.75 * 19.5

First, compute 247.75 * 19.5

Break it down:

247.75 * 20 = 4955
Minus 247.75 * 0.5 = 123.875
So 4955 - 123.875 = 4831.125

Now divide by 3: 4831.125 / 3 = 1610.375 in³

So 1610.4 in³ (rounded to nearest tenth)

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Problem 8: Square Pyramid

Given:
- Base is square, side = 30 ft → base area = 30×30=900 ft²
- Height = 17 ft (labeled vertically)

Volume = (1/3) * base area * height = (1/3) * 900 * 17

First, 900 / 3 = 300
Then 300 * 17 = 5100 ft³

Exactly 5100 ft³

Note: There’s also a "20 ft" labeled on the slant edge — but since height is given as 17 ft, we don’t need it. Probably red herring or for surface area, but question is volume.

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Now, compiling final answers:

Problem 5: ~323.5 ft³
Problem 6: ~1785.6 km³
Problem 7: 1610.4 in³
Problem 8: 5100 ft³

But let me verify Problem 5 once more — is there a chance that 12.8 is the height? In some diagrams, labels might be ambiguous.

Looking back at user’s description: “12.8 ft” is written next to the slanted edge, and there’s a separate dashed line for height with right angle — so definitely slant height. Our calculation is correct.

Perhaps they want exact form? But unlikely.

Another way: maybe use 12.8 as height? Then volume = (1/3)*81*12.8 = 27*12.8 = 345.6 — but that would ignore the geometry shown.

I think 323.5 is correct.

But let me check online or standard practice — no, we must follow diagram.

Perhaps the 12.8 is the apothem? No, it’s labeled on the edge.

I’ll stick with 323.5.

Final Answers:

5. 323.5 ft³
6. 1785.6 km³
7. 1610.4 in³
8. 5100 ft³

But for Problem 6, let me use more precise π.

V = (1/3) * π * (8.5)^2 * 23.6 = (1/3) * π * 72.25 * 23.6

72.25 * 23.6 = 1705.1 as before

1705.1 * π = 1705.1 * 3.141592653589793 = let's compute:

1705.1 * 3.141592653589793

First, 1705.1 * 3 = 5115.3

1705.1 * 0.141592653589793 = ?

Compute 1705.1 * 0.14 = 238.714

1705.1 * 0.001592653589793 ≈ 1705.1 * 0.00159265 ≈

1705.1 * 0.001 = 1.7051
1705.1 * 0.0005 = 0.85255
1705.1 * 0.00009265 ≈ 1705.1 * 0.00009 = 0.153459, plus 1705.1*0.00000265≈0.004518 → total ≈ 0.157977

So total for 0.00159265 ≈ 1.7051 + 0.85255 = 2.55765 + 0.157977 ≈ 2.715627

Thus 0.14159265 part ≈ 238.714 + 2.715627 = 241.429627

Total π part = 5115.3 + 241.429627 = 5356.729627

Divide by 3: 1785.576542 → so 1785.6 km³ is fine.

Similarly, others are good.

Final Answer:
5. 323.5 ft³
6. 1785.6 km³
7. 1610.4 in³
8. 5100 ft³