Mix Examples - Surface Areas and Volumes Questions in English

(A) Given: Diameter of the base $(d)$ = $7 \,cm$.
Radius $(r)$ = $d / 2 = 7 / 2 = 3.5 \,cm$.
Height $(h)$ = $12 \,cm$.
The formula for the curved surface area of a cylinder is $2 \pi r h$.
Curved Surface Area = $2 \times (22 / 7) \times 3.5 \times 12$.
Curved Surface Area = $2 \times (22 / 7) \times (7 / 2) \times 12$.
Curved Surface Area = $22 \times 12 = 264 \,cm^2$.

(B) The total surface area of a hemisphere is given by the formula $TSA = 3 \pi r^2$.
Given the radius $r = 42 \, cm$.
Substituting the values,we get:
$TSA = 3 \times \frac{22}{7} \times 42 \times 42$
$TSA = 3 \times 22 \times 6 \times 42$
$TSA = 66 \times 252$
$TSA = 16632 \, cm^2$.

(C) The total surface area of a cone is given by the formula: $TSA = \pi r(r + l)$,where $r$ is the radius and $l$ is the slant height.
Given: $r = 10 \, cm$,$l = 25 \, cm$.
Using $\pi \approx \frac{22}{7}$ is not ideal here,let us check if $\pi \approx 3.14$ is intended or if the result is a multiple of $\pi$.
Calculation: $TSA = \pi \times 10 \times (10 + 25) = \pi \times 10 \times 35 = 350\pi$.
If we take $\pi \approx 3.14$,$TSA = 350 \times 3.14 = 1099 \approx 1100 \, cm^2$.
Thus,the correct option is $C$.

(D) The lateral surface area of a cuboid is given by the formula: $2h(l + b)$.
Given: length $(l)$ = $30 \, cm$,breadth $(b)$ = $20 \, cm$,and height $(h)$ = $8 \, cm$.
Substituting the values into the formula:
Lateral Surface Area = $2 \times 8 \times (30 + 20)$
Lateral Surface Area = $16 \times 50$
Lateral Surface Area = $800 \, cm^2$.
Therefore,the correct option is $D$.

(A) The volume $V$ of a sphere is given by the formula $V = \frac{4}{3} \pi r^{3}$.
Given,radius $r = 2.1 \, m$.
Substituting the value of $r$ in the formula:
$V = \frac{4}{3} \times \frac{22}{7} \times (2.1)^{3}$
$V = \frac{4}{3} \times \frac{22}{7} \times 2.1 \times 2.1 \times 2.1$
$V = 4 \times 22 \times 0.1 \times 2.1 \times 2.1$
$V = 88 \times 0.1 \times 4.41$
$V = 8.8 \times 4.41 = 38.808 \, m^{3}$.

(B) Given: Radius $(r)$ = $7 \, cm$,Height $(h)$ = $24 \, cm$.
First,find the slant height $(l)$ using the formula: $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \, cm$.
The curved surface area $(CSA)$ of a cone is given by the formula: $CSA = \pi rl$.
$CSA = \frac{22}{7} \times 7 \times 25$.
$CSA = 22 \times 25 = 550 \, cm^2$.

(C) The surface area of a sphere is given by the formula $A = 4 \pi r^2$,where $r$ is the radius.
Given $A = 2464 \, cm^2$ and taking $\pi = \frac{22}{7}$:
$4 \times \frac{22}{7} \times r^2 = 2464$
$r^2 = \frac{2464 \times 7}{4 \times 22}$
$r^2 = \frac{2464 \times 7}{88}$
$r^2 = 28 \times 7 = 196$
$r = \sqrt{196} = 14 \, cm$
The diameter $d = 2r = 2 \times 14 = 28 \, cm$.

(D) Given,the diameter of the hemisphere $d = 42\, cm$.
Therefore,the radius $r = \frac{d}{2} = \frac{42}{2} = 21\, cm$.
The formula for the curved surface area $(CSA)$ of a hemisphere is $2\pi r^2$.
Substituting the values,we get:
$CSA = 2 \times \frac{22}{7} \times (21)^2$
$CSA = 2 \times \frac{22}{7} \times 21 \times 21$
$CSA = 2 \times 22 \times 3 \times 21$
$CSA = 44 \times 63 = 2772\, cm^2$.
Thus,the curved surface area of the hemisphere is $2772\, cm^2$.

(A) The volume of a cube with edge $a = 15\, cm$ is given by $V_{cube} = a^3 = 15^3 = 3375\, cm^3$.
The volume of a cuboid is given by $V_{cuboid} = \text{length} \times \text{breadth} \times \text{height} = l \times b \times h$.
Given $l = 25\, cm$ and $b = 15\, cm$,and $V_{cuboid} = V_{cube}$,we have:
$25 \times 15 \times h = 3375$
$375 \times h = 3375$
$h = \frac{3375}{375} = 9\, cm$.
Thus,the height of the cuboid is $9\, cm$.

(B) Given that the radius of the cylinder $(r_c)$ and the radius of the sphere $(r_s)$ are equal,so $r_c = r_s = 9 \, cm$.
Let the height of the cylinder be $h$.
The volume of the cylinder is given by $V_c = \pi r_c^2 h$.
The volume of the sphere is given by $V_s = \frac{4}{3} \pi r_s^3$.
According to the problem,the volumes are equal: $V_c = V_s$.
Substituting the formulas: $\pi r_c^2 h = \frac{4}{3} \pi r_s^3$.
Since $r_c = r_s = 9$,we can cancel $\pi$ and $r^2$ from both sides:
$h = \frac{4}{3} r_s = \frac{4}{3} \times 9$.
$h = 4 \times 3 = 12 \, cm$.
Thus,the height of the cylinder is $12 \, cm$.