Directions: In the questions below consists of a question and two stat...
Understanding the Problem
To find the total surface area of a cylinder, we need its height (h) and radius (r). The total surface area (TSA) is calculated using the formula:
TSA = 2πr(h + r)
Now, let's analyze the provided statements.
Statement I: Ratio of Curved Surface Area to Total Surface Area
- The ratio of the curved surface area (CSA) to the total surface area (TSA) is given as 2:3.
- This means CSA = (2/3) * TSA.
To express this relationship mathematically:
- CSA = 2πrh
- TSA = 2πr(h + r)
From this ratio, we can derive a relationship between CSA and TSA, but we still need the individual dimensions (h and r) to compute TSA directly. Thus, Statement I alone is not sufficient.
Statement II: Perimeter of the Rectangle from Curved Surface
- When the curved surface of the cylinder is cut and opened into a rectangle, the perimeter is given as 116 cm.
- The perimeter of the rectangle formed is calculated as P = 2(l + b), where l = height (h) and b = circumference (2πr).
- Therefore, we can establish the equation: 2(h + 2πr) = 116.
From this equation, we can derive the values of h and r, but we still need the relationship established in Statement I to find the total surface area accurately. Therefore, Statement II alone is also not sufficient.
Conclusion
Since neither statement alone is sufficient to find the total surface area of the cylinder, we require both statements together to extract the necessary dimensions and calculate TSA accurately.
Thus, the correct answer is option D: both statements together are necessary to answer the question.
Directions: In the questions below consists of a question and two stat...
Statement I:
No data about the length.
So, statement I alone is not sufficient.
Statement II:
2(2πr + h) = 116
But we cannot find the value of r or h.
So, statement II alone is not sufficient.
Using both the statements:
2πrh : 2πr(r + h) = 2 : 3
h : r = 2 : 1
2(2πr + h) = 116
Solving the above equations, we get r = 7 and h = 14
So, both the statements are together necessary.
Hence, option D is correct.