Question

1. suppose you have one pyramid with a base that is 10 meters long and 10 meters wide with a slant that is 15 meters. now, if the length of the base is increased to 15 meters, what is the difference in the surface area between the new pyramid and the old pyramid?

525 meters squared.
400 meters squared.
600 meters squared.
125 meters squared.

Explanation:

Step1: Recall Surface Area of Square Pyramid

The surface area \( SA \) of a square pyramid is \( SA = \text{Base Area} + \text{Lateral Surface Area} \). The base area is \( s^2 \) (where \( s \) is the side length of the square base), and the lateral surface area is \( 2sl \) (where \( l \) is the slant height, and there are 4 triangular faces, but for a square base with side \( s \), each triangular face has area \( \frac{1}{2}sl \), so 4 of them give \( 4\times\frac{1}{2}sl = 2sl \)). Wait, actually, for a square base with side \( s \), lateral surface area (LSA) is \( 2s \times l \) (since two pairs of opposite triangles, each pair with base \( s \) and slant height \( l \), so total LSA is \( 2\times(\frac{1}{2}sl) + 2\times(\frac{1}{2}sl) = 2sl \))? Wait, no, actually, for a square pyramid, the lateral surface area is \( 4\times\frac{1}{2}sl = 2sl \), where \( s \) is the side length of the base and \( l \) is the slant height. The total surface area is \( s^2 + 2sl \) (base area + lateral surface area).

Step2: Calculate Old Pyramid Surface Area

Old pyramid: base side \( s_1 = 10 \) m, slant height \( l = 15 \) m.
Base area \( A_{b1} = s_1^2 = 10^2 = 100 \) m².
Lateral surface area \( LSA_1 = 2\times s_1\times l = 2\times10\times15 = 300 \) m².
Total surface area \( SA_1 = A_{b1} + LSA_1 = 100 + 300 = 400 \) m²? Wait, no, wait: wait, the lateral surface area for a square pyramid is actually \( 4\times\frac{1}{2}sl = 2sl \), yes. But wait, maybe I made a mistake. Wait, no, let's recheck: each triangular face has area \( \frac{1}{2} \times \text{base} \times \text{slant height} \). The base of each triangular face is the side of the square base, so for a square base with side \( s \), there are 4 triangular faces, each with base \( s \) and slant height \( l \). So LSA is \( 4\times\frac{1}{2}sl = 2sl \). So for old pyramid: \( s = 10 \), \( l = 15 \). So LSA = \( 2\times10\times15 = 300 \), base area = \( 10^2 = 100 \), so total SA = 100 + 300 = 400? Wait, but let's check with another approach. Wait, maybe the problem is that the base is a square (10x10), so it's a square pyramid. Now, when we increase the length of the base to 15 meters, is the base still a square? Wait, the original base is 10m long and 10m wide, so it's a square. If we increase the length to 15m, is the width also increased? Wait, the problem says "the length of the base is increased to 15 meters" – maybe the base becomes a rectangle? Wait, original base: length 10, width 10 (square). New base: length 15, width 10? Wait, the problem says "the length of the base is increased to 15 meters" – maybe the base is a rectangle, originally 10x10 (square), now 15x10 (rectangle). Then, the lateral surface area would be different. Oh! Maybe I misinterpreted the base. The original base is 10m long and 10m wide, so it's a square (length = width). If we increase the length to 15m, what about the width? The problem says "the length of the base is increased to 15 meters" – maybe the width remains 10m? So the base is now a rectangle with length 15m and width 10m. Then, the lateral surface area would consist of two pairs of rectangular faces? No, wait, a pyramid with a rectangular base is a rectangular pyramid. The lateral surface area of a rectangular pyramid is \( \frac{1}{2} \times (2l + 2w) \times l_{slant} \), where \( l \) is length, \( w \) is width, and \( l_{slant} \) is the slant height. Wait, no, for a rectangular pyramid, there are two triangular faces with base \( l \) and height (slant height) \( l_1 \), and two triangular faces with base \( w \) and height (slant height) \( l_2 \). But the problem says "a slant that is 15 meters" – maybe the slant height is the same for all faces? Wait, the original pyramid has a base that is 10x10 (square), so it's a square pyramid, slant height 15m. When we increase the length to 15m, the base becomes 15x10 (rectangle), and we need to assume that the slant height for the new length (15m) and the original width (10m) – but maybe the slant height remains the same? Wait, the problem says "a slant that is 15 meters" – maybe the slant height (the height of the triangular faces) is 15m. Let's re-express:

Original pyramid: square base, side 10m, slant height (height of triangular faces) 15m. Total surface area: base area (10x10) + 4 triangular faces (each 10x15/2). So each triangular face area: \( \frac{1}{2} \times 10 \times 15 = 75 \) m². Four of them: 4×75=300 m². Base area: 100 m². Total SA: 100+300=400 m².

New pyramid: base is 15m long (length) and 10m wide (width). So the base is a rectangle, 15x10. Now, the lateral faces: two faces with base 15m and height (slant height) 15m, and two faces with base 10m and height (slant height) 15m? Wait, no, if the original pyramid had a square base, and we increase the length, maybe the width remains 10m, so the base is now a rectangle. Then, the lateral surface area would be: two triangles with base 15m, height 15m, and two triangles with base 10m, height 15m. Wait, but in the original pyramid, all four triangles had base 10m. Now, two triangles have base 15m, two have base 10m.

So original LSA: 4*(1/2)*10*15 = 300 m².

New LSA: 2*(1/2)*15*15 + 2*(1/2)*10*15 = (15*15) + (10*15) = 225 + 150 = 375 m².

Base area new: 15*10 = 150 m².

Total SA new: 150 + 375 = 525 m².

Difference: 525 - 400 = 125? No, wait, that's not matching. Wait, maybe the base remains a square? No, the problem says "the length of the base is increased to 15 meters" – length and width were both 10, now length is 15, width remains 10? Or is the base still a square, so length and width both become 15? Wait, the problem says "a base that is 10 meters long and 10 meters wide" – so length = width = 10. Then "the length of the base is increased to 15 meters" – maybe width remains 10, so base is 15x10. But maybe the problem is that the base is a square, and when length is increased, width is also increased? No, the problem says "the length of the base is increased to 15 meters" – so length becomes 15, width remains 10? Wait, this is confusing. Wait, let's check the answer options. The options are 525, 400, 600, 125.

Wait, maybe the original surface area is calculated as base area (10x10) + lateral surface area (4*(10*15/2)) = 100 + 300 = 400. Then the new base is 15x15 (square), so length and width both 15? Wait, the problem says "the length of the base is increased to 15 meters" – maybe width is also increased to 15? But it says "length" – maybe it's a square pyramid, so length = width. So original: 10x10, new: 15x15. Then:

Old SA: 10² + 4*(10*15/2) = 100 + 300 = 400.

New SA: 15² + 4*(15*15/2) = 225 + 450 = 675. Difference: 675 - 400 = 275. Not an option.

Alternatively, maybe the lateral surface area is only the area of the two longer sides? Wait, no. Wait, maybe the problem is that the base is a square, and when we increase the length (side) to 15, the surface area difference is the difference in lateral surface area (since base area difference is 15² - 10² = 225 - 100 = 125, and lateral surface area difference is 2*15*15 - 2*10*15 = 450 - 300 = 150, total difference 125 + 150 = 275. No.

Wait, maybe the problem is considering only the lateral surface area? Wait, the question says "surface area" – but maybe it's the lateral surface area. Let's check:

Old lateral surface area (LSA) for square base 10m: 4*(10*15/2) = 300.

New LSA for rectangular base 15x10: two faces with base 15, two with base 10. So LSA = 2*(15*15/2) + 2*(10*15/2) = 225 + 150 = 375. Difference: 375 - 300 = 75. Not an option.

Wait, maybe the original pyramid is a square pyramid, and when we increase the length (side) to 15, the slant height is still 15, and the surface area difference is (15² + 2*15*15) - (10² + 2*10*15) = (225 + 450) - (100 + 300) = 675 - 400 = 275. No.

Wait, the answer options include 125. Let's think differently. Maybe the base area is 10x10, and when length is increased to 15, the base area becomes 15x10, so base area difference is 15*10 - 10*10 = 50. Lateral surface area: original LSA is 2*10*15 = 300 (for square, 4 triangles: 2*10*15). New LSA: for rectangle 15x10, LSA is (15*15) + (10*15) = 225 + 150 = 375? Wait, no, 2*(15*15/2) + 2*(10*15/2) = 225 + 150 = 375. So total surface area difference: (15*10 + 375) - (10*10 + 300) = (150 + 375) - (100 + 300) = 525 - 400 = 125. Ah! That's one of the options (125 meters squared). Let's verify:

Old SA: base (10x10=100) + LSA (4*(10*15/2)=300) → 400.

New SA: base (15x10=150) + LSA (2*(15*15/2) + 2*(10*15/2) = 225 + 150 = 375) → 150 + 375 = 525.

Difference: 525 - 400 = 125. Yes! So that's the answer.

So the key was realizing that the base becomes a rectangle (15x10) instead of a square, so the lateral surface area has two faces with base 15 and two with base 10, each with slant height 15. Then calculating the difference in total surface area.

Step3: Calculate New Pyramid Surface Area

New pyramid: base length \( l = 15 \) m, base width \( w = 10 \) m (width remains 10m), slant height \( l = 15 \) m.
Base area \( A_{b2} = l \times w = 15 \times 10 = 150 \) m².
Lateral surface area: two triangular faces with base \( l = 15 \) m, area each \( \frac{1}{2} \times 15 \times 15 = 112.5 \) m², so two of them: \( 2 \times 112.5 = 225 \) m². Two triangular faces with base \( w = 10 \) m, area each \( \frac{1}{2} \times 10 \times 15 = 75 \) m², so two of them: \( 2 \times 75 = 150 \) m². Total LSA \( LSA_2 = 225 + 150 = 375 \) m².
Total surface area \( SA_2 = A_{b2} + LSA_2 = 150 + 375 = 525 \) m².

Step4: Calculate Difference

Difference \( = SA_2 - SA_1 = 525 - 400 = 125 \) m².

Answer:

125 meters squared.