3D Objects
A dessert can be modelled as a right-cone of radius 3 cm and height 12 cm and a scoop of ice-cream in the shape of a sphere of radius 3 cm. Find the total volume of the ice-cream and cone
226 cm^3
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he point (2a, a, 5a) is twice as far from the origin as the point (−4, 1, 7). Find the positive value of a.
a=2.97
These 3D shapes have one curved surface and two circular bases. What are they?
Cylinders
Pyramid ABCDE is square based The length of the side of the square is 12cm. The perpendicular height of the pyramid is 8cm.
Find the volume of the pyramid.
384 cm 3
Find the exact distance between points A (–1,2,3) and B (–1,–2,4)
AB= 4.12 OR Radical 17
What is the term for a 3D shape with a curved surface and no vertices or edges?
Sphere
Find the exact volume of the cone with base radius 3cm and perpendicular height 7cm .
V=21 pie cm^3
A(1, 4) and C(5, 10) are opposite vertices of a square ABCD.
a. Find the midpoint of AC.
b. Find the coordinates of B and D
a. (3,7)
b.(6,5) and (0,9)
What geometric property do points that lie on the midpoint of a line segment share?
Points that lie on the midpoint of a line segment divide the segment into two equal parts
Find the surface area of the hemisphere with a diameter of 10cm
A = 236 CM^2
Hodder Worked Example 5.1
The midpoint of the line segment connecting the points P(–4, a, 1) and Q(b, 1, 8) is M(8, 2, c). Find the values of a, b and c.
a=3,b=20,c=4.5
What is the name of a 3D shape that has 5 faces, 5 vertices, and 8 edges?
Pyramid
Volume
A small toy is made up of a cuboid with a right pyramid, if the volume is 150 cm^3, what is h ?
Given: In Cuboid
Length = 5 cm
Width = 6cm
Height = 4 cm
V of triangle= 1/3 (Ah)
V of cuboid = l x w x h
V total = 1/3 (Ah) + (l x w x h)
1/3(5 x 6)(h) + (5x6x4)
150= 10h + 120
30 = 10h
h=3
An empty room is 3m by 4m by 5m. A spider and a fly start in one corner of the room
a) The fly travels to the furthest corner of the room by flying and stops there. What is the shortest distance it can travel to get there?
b The spider then chases after the fly by crawling along any wall. What is the shortest distance it can travel to get to the fly’s final position?
a= 7.07 m
b=8.60 m
Explain how the distance formula is derived from the Pythagorean theorem in 2D space
The distance formula is derived by considering the lengths of the horizontal and vertical legs of a right triangle formed by the two points and using the Pythagorean theorem to find the hypotenuse.