The diagram below, not drawn to scale, shows a cuboid with length 13 cm, width 4 cm
and height h cm.
a. State, in terms of h, the area of the shaded face of the cuboid.
b. Write an expression, in terms of h, for the volume of the cuboid
c. If the volume of the cuboid is 286 cm 3 , calculate the height, h, of the cuboid.
d. Calculate the total surface area of the cuboid.
Sorry, I cannot provide a diagram as I am a language model and do not have the ability to display images.
To solve this problem, we need to understand the basic formulas related to cuboids.
A cuboid is a three-dimensional shape with six rectangular faces. The lengths of the edges are represented as length, width, and height. Let's proceed to solve the given questions step by step.
a. To find the area of the shaded face of the cuboid, we need to determine which face is shaded. From the given information, it is not specified which face is shaded. Therefore, the answer will be dependent on the height (h) of the cuboid.
b. The volume of a cuboid is given by the formula:
Volume = length * width * height
In this case, the expression in terms of height (h) for the volume of the cuboid will be:
Volume = 13 * 4 * h
= 52h cm³
c. Given the volume of the cuboid as 286 cm³, we can now solve for the height (h). The volume formula is already given as:
52h = 286
To find h, we divide both sides by 52:
h = 286 / 52
= 5.5 cm
Therefore, the height of the cuboid is 5.5 cm.
d. The total surface area of a cuboid is given by the formula:
Surface Area = 2lw + 2lh + 2wh
Using the given values, we can substitute the length (l), width (w), and height (h) to calculate the total surface area:
Surface Area = 2(13*4) + 2(13*h) + 2(4*h)
= 104 + 26h + 8h
= 104 + 34h cm²
And that is the expression for the total surface area of the cuboid in terms of the height (h).
a. The area of the shaded face of the cuboid can be found by determining the dimensions of the face that is shaded and calculating its area. In this case, the shaded face is a rectangle with dimensions 13 cm (length) and h cm (height). Therefore, the area of the shaded face is given by the formula:
Area = Length × Height = 13 cm × h cm = 13h square cm.
b. The volume of a cuboid is determined by multiplying its length, width, and height. In this case, the length is 13 cm, the width is 4 cm, and the height is h cm. Therefore, the volume of the cuboid is given by the formula:
Volume = Length × Width × Height = 13 cm × 4 cm × h cm = 52h cubic cm.
c. Given that the volume of the cuboid is 286 cubic cm, we can set up an equation to solve for the height, h:
Volume = 52h cubic cm
286 = 52h
To solve for h, divide both sides of the equation by 52:
h = 286 / 52 = 5.50 cm
Therefore, the height of the cuboid is 5.50 cm.
d. The total surface area of a cuboid consists of the sum of the areas of all its faces. In this case, the cuboid has 6 faces: 2 faces with dimensions 13 cm (length) × 4 cm (width), 2 faces with dimensions 13 cm (length) × h cm (height), and 2 faces with dimensions 4 cm (width) × h cm (height).
The surface area can be calculated using the formula:
Surface Area = 2 × (Length × Width + Length × Height + Width × Height)
Substituting the values:
Surface Area = 2 × (13 cm × 4 cm + 13 cm × h cm + 4 cm × h cm)
Surface Area = 2 × (52 cm² + 13h cm² + 4h cm²)
Surface Area = 2 × (52 + 17h + 4h) cm²
Surface Area = 2 × (52 + 21h) cm²
Surface Area = 104 + 42h cm²
Therefore, the total surface area of the cuboid is 104 + 42h square cm.