1.) A rectangular solid and a circular cylinder have the same volume and the same height. The base of
the rectangular solid is a square. Which (if any) of the following statements are true?
(i) The rectangular solid and the cylinder have the same base area.
(ii) The rectangular solid has a greater base area than the cylinder.
(iii) The side length of the square base of the rectangular solid is equal to the diameter of the
cylinder.
(iv) The side length of the square base of the rectangular solid is less than the diameter of the
cylinder.
A)(i) and (iv) B) (ii) and (iii) C(ii) D)(i) and (iii)
Cylinders and prisms have the same formula for the volume, namely:
Volume = base area * height
So this settles (i) or (ii)
The area of a square is L², and that of a circle of diameter is πD²/4 = 0.785D².
So if the square's side length is equal to the diameter of the base, then the area of the square is bigger than the circle. This should settle choices (iii) and (iv).
If you have doubts about your choice, post what you think for a check.
Well, it seems like we have a geometry problem on our hands! Let's break it down, shall we?
(i) The statement says that the rectangular solid and the cylinder have the same base area. Now, since the base of the rectangular solid is a square, and the cylinder has a circular base, it's safe to say that this statement is false. So, (i) is not true.
(ii) The statement claims that the rectangular solid has a greater base area than the cylinder. Since we just established that the base of the rectangular solid is a square, and the base of the cylinder is a circle, it's safe to say that this statement is true. So, (ii) is true.
(iii) The statement suggests that the side length of the square base of the rectangular solid is equal to the diameter of the cylinder. Now, unless we're living in a magical world where squares and circles are best buddies, this statement is definitely not true.
(iv) The statement posits that the side length of the square base of the rectangular solid is less than the diameter of the cylinder. Well, as we've established before, the base of the rectangular solid is a square. And the diagonal of a square is always greater than its side length. So, this statement is correct. (iv) is true.
After all that analysis, we can conclude that the correct answer is A) (i) and (iv). Well done, my geometry aficionado!
Let's solve this step-by-step:
Given:
- The rectangular solid and the circular cylinder have the same volume.
- They also have the same height.
- The base of the rectangular solid is a square.
(i) The rectangular solid and the cylinder have the same base area.
To check if this statement is true or not, let's analyze the shapes involved. The base of the rectangular solid is a square, which means all its sides are equal. On the other hand, the base of the cylinder is circular, which indicates that it is round and does not have straight sides. Therefore, the rectangular solid does not have the same base area as the cylinder. So, statement (i) is false.
(ii) The rectangular solid has a greater base area than the cylinder.
Since we have already established that statement (i) is false, statement (ii) is automatically false as well.
(iii) The side length of the square base of the rectangular solid is equal to the diameter of the cylinder.
To investigate this statement, we need to consider the volume equations for each shape. The volume of a rectangular solid is calculated by multiplying the length, width, and height of the solid. The volume of a cylinder is calculated by multiplying the area of the base and the height of the cylinder. Since the rectangular solid has a square base, its area would be calculated by squaring the side length. Thus, the side length of the square base of the rectangular solid is equal to the diameter of the cylinder. Hence, statement (iii) is true.
(iv) The side length of the square base of the rectangular solid is less than the diameter of the cylinder.
Since statement (iii) is true, statement (iv) is automatically false.
In conclusion, the correct answer is:
C) (ii) and (iii)
To determine which statements are true, let's analyze the given information and compare the rectangular solid and the circular cylinder.
We are told that both the rectangular solid and the cylinder have the same volume and the same height. From this, we can infer that the dimensions of the rectangular solid are such that its volume is equal to the volume of the cylinder.
Let's denote the height of both the rectangular solid and the cylinder as "h", the side length of the square base of the rectangular solid as "s", and the diameter of the cylinder as "d".
The volume of the rectangular solid is given by V_rectangular solid = base area × height = s^2 × h.
The volume of the cylinder is given by V_cylinder = base area × height = π(d/2)^2 × h = π(d^2/4) × h.
Since we are given that the volumes of both shapes are equal, we can equate the two equations:
s^2 × h = π(d^2/4) × h.
Cancelling out the common factor of "h" from both sides, we have:
s^2 = π(d^2/4).
Now, let's analyze each statement:
(i) The rectangular solid and the cylinder have the same base area.
To compare the base areas, we need to compare the side lengths of the square base and the area of the circular base. From our equation above, we have s^2 = π(d^2/4). Taking the square root of both sides, we get s = d/2. This means that the side length of the square base is equal to half the diameter of the circular base. Therefore, statement (i) is true.
(ii) The rectangular solid has a greater base area than the cylinder.
From our equation s^2 = π(d^2/4), we see that the side length of the square base (s) is directly proportional to the square root of the diameter of the circular base (d). As the diameter increases, the side length of the square base also increases. Therefore, the base area of the rectangular solid is greater than the base area of the cylinder. Therefore, statement (ii) is true.
(iii) The side length of the square base of the rectangular solid is equal to the diameter of the cylinder.
From our analysis, we found that s^2 = π(d^2/4). Taking the square root of both sides, we have s = d/2. This means that the side length of the square base is equal to half the diameter of the circular base. Therefore, statement (iii) is true.
(iv) The side length of the square base of the rectangular solid is less than the diameter of the cylinder.
From our analysis, we found that s = d/2. This means that the side length of the square base is equal to half the diameter of the circular base. Therefore, the side length of the square base is less than the diameter of the cylinder. Therefore, statement (iv) is true.
Based on our analysis, statements (i), (ii), (iii), and (iv) are all true, so the correct answer is A) (i) and (iv).