Math, asked by HarryLilyPotter, 1 month ago

COMPREHENSION-TYPE QUESTIONS

A tower has the following shape: a truncated right circular cone (one with radii 2R (the lower base) and R (the upper base)), and the height R bears a right circular cylinder whose radius is R, the height being 2R. Finally a semisphere of radius R is mounted on the cylinder. Suppose that the
cross-sectional area S of the tower is given by f(x), where
x is the distance of the cross-section from the lower base of the cone.

Q1) The domain of the function f(x) is
(a) [0, 4R]
(b) [R, 4R]

Q2) For 0<= x <= R, the function f(x) is given by
(a) π(2R-x)²
(b) π(R-x)²

Q3) The range of f(x) is
(a) [0 , 4πR²]
(b) (0, R]

Q4) The function f(x) is
(a) one-one on (R, 2R]
(b) one-one on [R, 3R]
(c) one-one on [0, 4R)
(d) one-one on [0, R] U [3R, 4R]

Answers

Answered by Afreenakbar

Answer:

1) The domain of the function f(x) is (a) [0, 4R].

2) For 0 <= x <= R, the function f(x) is given by (b) π(R - x)².

3) The range of f(x) is (a) [0, 4πR²].

4) The function f(x) is (d) one-one on [0, R] U [3R, 4R].

Step-by-step explanation:

Let's examine each question in turn in order to provide answers regarding the tower's design and the function f(x):

1) This is the scope of the function f(x):

The function's acceptable input values are represented by the domain. In this instance, the cross-sectional area of the tower at a distance x from the lower base of the cone is described by the function f(x). The correct range of x would be from 0 to the height of the entire tower, which is 4R, since the tower extends from the lower base of the cone to the top. Therefore, (a) [0, 4R] is the best choice.

2) For 0 <= x <= R, the function f(x) is given by:

The truncated cone part is all that remains of the tower in this range. With R being the radius of the larger base and r being the radius of the smaller base, the formula π(R + r)(R - r) can be used to calculate the cross-sectional area of a truncated cone. In this case, R = 2R and r = R, so the formula becomes π(2R + R)(2R - R), which simplifies to π(R - x)². Therefore, the correct option is (b) π(R - x)².

3) The range of f(x) is:

The possible output values for the function are represented by the range. F(x) cannot be negative because it indicates the tower's cross-sectional area. Additionally, near the base of the semisphere, which has an area of πR², the cross-sectional area is at its highest value.Therefore, the range of f(x) would be from 0 to πR². Hence, the correct option is (a) [0, 4πR²].

4) f(x) is the function that is:

If various input values result in different output values, then the function f(x) is one-to-one (injective). If not, then it is not. The cross-sectional area of the tower is here represented by the function f(x), and as x rises, so does the area of the cross-section. But throughout the entire domain, the function is not one-to-one. The intervals [0, R] and [3R, 4R] have a one-to-one relationship. The right answer is (d) one-one on [0, R] U [3R, 4R].