Answer:
[tex]Volume\text{ of the box = 3847.60 cm}^3[/tex]
Explanation:
Given:
The box is in the shape of a semisphere
The edge of the cubic cheese = 10cm
To find:
the volume of the box
The box is said to be a semisphere. This is also known as the hemisphere.
To get the volume of the box, we will apply the volume of a hemisphere.
The volume is given as:
[tex]Volume\text{ of a hemisphere = }\frac{2}{3}\pi r^3[/tex]
The 4 top vertices start from the cover of the box (hemisphere).
From the diagram, radius = AC
AB is an edge = 10
We need to get the distance BD using pythagoras theorem:
Hypotenuse² = opposite² + adjacent²
opposite = 10, adjacent = 10
BD² = 10² + 10²
BD² = 200
BD = √200 = √(100×2)
BD = 10√2
Next, we will find BC:
BD = BC + CD
BC = CD
BD = 2BC
BC = BD/2
BC = (10√2)/2
To get the radius, we will use Pythagoras theorem on triangle ABC
AB = opposite = 10
BC = adjacent = (10√2)/2
AC = hypotenuse = radius
Substitute the values into the Pythagoras formula:
[tex]\begin{gathered} radius²\text{ = 10^^b2 + \lparen}\frac{10\sqrt{2}}{2})^2 \\ \\ radius²\text{ = 100 + }\frac{10\sqrt{2}\text{ }\times10\sqrt{2}}{2^2} \\ \\ radius^2\text{ = 100 + }\frac{100(2)}{4}\text{ = 100 + 50} \\ \\ radius^2\text{ = 150} \\ \\ radius\text{ = }\sqrt{150}\text{ = 12.2474} \end{gathered}[/tex][tex]\begin{gathered} Volume\text{ of a hemisphere = }\frac{2}{3}\pi r^3 \\ \\ let\text{ }\pi\text{ = the value on a calculator} \\ r\text{ = 12.2474} \\ \\ Volume\text{ of the hemisphere = }\frac{2}{3}\times\pi\times(12.2474)^3 \\ \\ Volume\text{ of the hemisphere = 3847.60 cm}^3 \\ \\ Volume\text{ of the box= 3847.60 cm}^3 \end{gathered}[/tex]
