Question

lateral and total surface area of rectangular prisms.
ill correctly solve problems involving the lateral and
at least 4 of 5 questions.

1. maria wants to recycle a gift box, so she

carefully cuts it open along the edges and
unravels it into a flat shape.

image of a rectangular prism net with dimensions: 4 cm, 2 cm, 10 cm

what is the lateral surface area of the
gift box?
______ cm²

Explanation:

Step1: Recall lateral surface area formula for rectangular prism

The lateral surface area (LSA) of a rectangular prism is given by the formula \( LSA = 2h(l + w) \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height. From the net, we can identify the dimensions: length \( l = 10 \) cm, width \( w = 4 \) cm, height \( h = 2 \) cm.

Step2: Substitute values into the formula

First, calculate \( l + w \): \( 10 + 4 = 14 \) cm. Then, multiply by \( 2h \): \( 2\times2\times14 = 4\times14 = 56 \)? Wait, no, wait. Wait, actually, the lateral surface area can also be calculated as the sum of the areas of the four lateral faces. The lateral faces are the ones without the top and bottom. Looking at the net, the lateral faces: two faces with dimensions \( 10 \times 2 \) and two faces with dimensions \( 4 \times 2 \)? Wait, no, wait. Wait, the rectangular prism has length \( l = 10 \) cm, width \( w = 4 \) cm, height \( h = 2 \) cm. Wait, no, maybe I mixed up. Wait, the lateral surface area is the perimeter of the base times the height. The base is a rectangle with length \( 10 \) and width \( 4 \)? No, wait, no. Wait, the formula for lateral surface area of a rectangular prism is \( 2h(l + w) \), where \( l \) and \( w \) are the length and width of the base, and \( h \) is the height. Wait, but looking at the net, the height (the vertical side) is 2 cm? Wait, no, maybe the dimensions are: length \( l = 10 \) cm, width \( w = 4 \) cm, height \( h = 2 \) cm. Wait, no, let's look at the net again. The gray rectangles: two of them are \( 10 \times 4 \)? No, wait, the net has two rectangles of \( 10 \times 4 \) (the top and bottom? No, lateral faces. Wait, no, lateral surface area is the area of the sides excluding top and bottom. So the lateral faces: for a rectangular prism, the lateral faces are the four faces that are not the top and bottom. So if the base is \( l \times w \), then the lateral faces are two \( l \times h \) and two \( w \times h \). So from the net, the height \( h = 2 \) cm, length \( l = 10 \) cm, width \( w = 4 \) cm. So the lateral surface area would be \( 2(lh + wh) = 2h(l + w) \). Let's compute that: \( l = 10 \), \( w = 4 \), \( h = 2 \). So \( 2\times2\times(10 + 4) = 4\times14 = 56 \)? Wait, no, that can't be. Wait, maybe I got the dimensions wrong. Wait, looking at the net, the vertical sides: the small rectangles (the ones with 2 cm height) are 4 cm in length? Wait, no, the net has: the top and bottom are 4 cm by 2 cm? No, the gray rectangles are 10 cm by 4 cm? Wait, no, the first gray rectangle is 10 cm (height) by 4 cm (width). Then the adjacent white rectangles: one is 10 cm by 2 cm? Wait, no, maybe the correct dimensions are: length \( l = 10 \) cm, width \( w = 4 \) cm, height \( h = 2 \) cm. Wait, no, let's re-express. The lateral surface area is the sum of the areas of the four lateral faces. So two faces with area \( l \times h \) and two faces with area \( w \times h \). So if \( l = 10 \), \( w = 4 \), \( h = 2 \), then lateral surface area is \( 2\times(10\times2) + 2\times(4\times2) = 40 + 16 = 56 \)? Wait, but that seems low. Wait, no, maybe the length is 10, width is 2, height is 4? Wait, maybe I mixed up the dimensions. Let's look at the net again. The top and bottom squares (or rectangles) are 4 cm by 2 cm. The vertical rectangles (the gray ones) are 10 cm by 4 cm? No, the gray rectangles have height 10 cm and width 4 cm? Wait, no, the first gray rectangle is 10 cm (vertical) by 4 cm (horizontal). Then the white rectangles next to it: one is 10 cm (vertical) by 2 cm (horizontal)? Wait, no, the net is a rectangular prism net, so the dimensions are: length \( l = 10 \) cm, width \( w = 4 \) cm, height \( h = 2 \) cm. Wait, but then the lateral surface area is \( 2h(l + w) = 2\times2\times(10 + 4) = 4\times14 = 56 \). Wait, but let's check by adding the areas of the lateral faces. The lateral faces: in the net, the lateral faces are the four rectangles that are not the top and bottom (the white ones with 4x2? No, the top and bottom are the small rectangles (4x2) and the lateral faces are the ones with 10x2 and 10x4? Wait, no, I think I made a mistake. Wait, the correct formula for lateral surface area of a rectangular prism is \( 2(lh + wh) \), where \( l \) is length, \( w \) is width, \( h \) is height. Let's take \( l = 10 \), \( w = 4 \), \( h = 2 \). Then \( lh = 10\times2 = 20 \), \( wh = 4\times2 = 8 \). Then \( 2(20 + 8) = 2\times28 = 56 \). Wait, but let's look at the net. The lateral faces: two faces with area \( 10\times2 \) (the ones on the sides) and two faces with area \( 4\times2 \) (the ones on the top and bottom? No, top and bottom are the ones with 4x2, so lateral faces are the ones with 10x2 and 10x4? Wait, no, I'm confused. Wait, maybe the length is 10, width is 2, height is 4. Let's try that. Then \( l = 10 \), \( w = 2 \), \( h = 4 \). Then lateral surface area is \( 2(10\times4 + 2\times4) = 2(40 + 8) = 2\times48 = 96 \). No, that's not right. Wait, looking at the net, the gray rectangles are 10 cm in height and 4 cm in width. The white rectangles (the ones adjacent to the gray ones) are 10 cm in height and 2 cm in width? No, the white rectangles have height 10 cm and width 2 cm? Wait, no, the top and bottom rectangles (the small ones) are 4 cm in length and 2 cm in width. So the rectangular prism has length \( l = 10 \) cm, width \( w = 4 \) cm, height \( h = 2 \) cm. Wait, then the lateral surface area is the area of the four sides: two sides with area \( l \times h = 10 \times 2 = 20 \) and two sides with area \( w \times h = 4 \times 2 = 8 \). So total lateral surface area is \( 2\times20 + 2\times8 = 40 + 16 = 56 \). Wait, but that seems small. Wait, maybe I mixed up length and height. Wait, maybe the height is 10 cm, length is 4 cm, width is 2 cm. Then lateral surface area would be \( 2(4\times10 + 2\times10) = 2(40 + 20) = 120 \). Wait, that makes more sense. Let's re-examine the net. The gray rectangles are 10 cm (height) by 4 cm (length). The white rectangles (the lateral faces) are 10 cm (height) by 2 cm (width). So the length \( l = 4 \) cm, width \( w = 2 \) cm, height \( h = 10 \) cm. Then the lateral surface area is \( 2h(l + w) = 2\times10\times(4 + 2) = 20\times6 = 120 \) \( cm^2 \). Ah, that must be it. I had the height wrong. The height of the prism is 10 cm, length is 4 cm, width is 2 cm. So the lateral surface area is the perimeter of the base (which is \( 2(l + w) \)) times the height \( h \). The base is a rectangle with length 4 and width 2, so perimeter is \( 2(4 + 2) = 12 \) cm. Then multiply by height 10 cm: \( 12\times10 = 120 \) \( cm^2 \).

Step3: Verify with net

Looking at the net, the lateral faces are the four rectangles that are not the top and bottom (the small 4x2 rectangles). The lateral faces: two rectangles with dimensions 10x4 and two rectangles with dimensions 10x2. Wait, no, 10x4 and 10x2? Wait, no, the gray rectangles are 10x4, and the white rectangles are 10x2. So the area of the two gray lateral faces: \( 2\times(10\times4) = 80 \), and the area of the two white lateral faces: \( 2\times(10\times2) = 40 \). Then total lateral surface area: \( 80 + 40 = 120 \) \( cm^2 \). Yes, that matches. So the correct dimensions are length \( l = 4 \) cm, width \( w = 2 \) cm, height \( h = 10 \) cm. So lateral surface area is \( 2h(l + w) = 2\times10\times(4 + 2) = 120 \) \( cm^2 \).

Answer:

\( 120 \)