Chapter 7 Mensuration Class 10 Maharashtra Board TextBook Solution
Practice set 7.1 | Q 7 | Page 145
A cylinder and a cone have equal bases. The height of the cylinder is 3 cm and the area of its base is 100 cm2. The cone is placed upon the cylinder. The volume of the solid figure so formed is 500 cm3. Find the total height of the figure
Answer:-
The problem provides the height of the cylinder as 3 cm and the area of its base as 100 cm². Let’s denote the radius of the cylinder as “r” and the height of the cone as “H”.
We can use the formula for the area of a circle to find the radius of the cylinder:
A = πr² 100 = πr² r = 10/π cm
The volume of the solid figure formed by placing the cone on top of the cylinder is given as 500 cm³. We know that the volume of a cylinder is given by:
V_cylinder = πr²h
We can substitute the values of “r” and “h” into this formula to find the volume of the cylinder:
V_cylinder = π(10/π)²(3) V_cylinder = 300/π cm³
Let’s now consider the cone that is placed on top of the cylinder. The volume of a cone is given by:
V_cone = (1/3)πr²H
We can substitute the value of “r” into this formula and simplify it:
V_cone = (1/3)π(10/π)²H V_cone = (100/3π)H cm³
The total volume of the solid figure formed by placing the cone on top of the cylinder is the sum of the volumes of the cylinder and the cone:
V_total = V_cylinder + V_cone 500 = 300/π + (100/3π)H
Simplifying this equation, we get:
500 – 300/π = (100/3π)H H = (3/100)(500 – 300/π) H ≈ 6 cm
Therefore, the height of the cone is approximately 6 cm. The total height of the solid figure is the sum of the height of the cylinder and the height of the cone:
Total height = h + H = 3 + 6 = 9 cm
Thus, the total height of the figure is 9 cm.
Solution
$$text{Height of the cylinder,} h = 3 text{cm}$$
Let the radius of the cylinder be $r text{cm}$ and the height of the cone be $H text{cm}$.
Area of the base of cylinder $= 100 text{cm}^2$
$$therefore pi r^2 = 100 qquad dots(1)$$
The volume of the solid figure $= 500 text{cm}^3$
$$therefore text{Volume of the cylinder} + text{Volume of the cone} = 500 text{cm}^3$$
$$Rightarrow pi r^2 h + frac{1}{3}pi r^2 H = 500$$
$$Rightarrow pi r^2 left( h + frac{H}{3} right) = 500$$
$$Rightarrow 100left( 3 + frac{H}{3} right) = 500 quad left[text{Using} (1)right]$$
$$Rightarrow 3 + frac{H}{3} = frac{500}{100} = 5$$
$$Rightarrow frac{H}{3} = 5 – 3 = 2$$
$$Rightarrow H = 6 text{cm}$$
Therefore, total height of the figure $$= h + H = 3 + 6 = 9 text{cm}$$
Thus, the total height of the figure is $$9 text{cm}$$.
Chapter 7 Mensuration Textbook Solution 7.1
Practice set 7.1
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