Geometry & trigonometry Flashcards
A new dictionary in the shape of a rectangular prism will have a thickness of 3 inches. The volume of the dictionary will be
216in^3. What must be the area of the front cover, a face perpendicular to the thickness, in square inches?
The volume of a rectangular prism is:
V = lwh
We can choose h to be the thickness, and in doing so make the cover of the book be a rectangle of length l and width
w. The area of the cover is then:
A = lw
We can substitute A for lw in the volume equation to get: V = Ah
We are given the volume and the height (thickness). Therefore, we can solve for the area of the cover:
216in ^3 = A(3in)
216in^3/3 in = A
72in^2 = A
The correct answer is
72in^2
A multi-layer cake is in the shape of a right cylinder. The height of the cake is 20 centimeters, and its radius is 10cm. If each of the cake layers has a volume of approximately 1,250 cubic centimeters, then how many layers does the cake have?
V= πr^2 h
V= π10^2 20
V= 2000π
V = 6283.2
Number of layers: 6283.2 / 1250
Approximately 5 layers
The volume of a triangular prism is 160 cubic centimeters
[\left(\text{cm}^3\right)]. One of the rectangular faces of the triangular prism has a side of length
[8] centimeters
[(\text{cm})] and a side of length
[5\,\text{cm}], as shown. What is the vertical height,
[h], in centimeters, of the triangular face of the prism?
A. 5
B. 8
C. 12
D. 40
Choice B
V = (area of base)(5) = 160
1/2 bh
32 = 1/2(8)h
h = 8
A tent forms an equilateral triangular prism, with both triangular faces exposed. Each side of the triangular face has a length of
[196] centimeters, and the tent is 250cm long. What is the height, in centimeters, of the tent?
A. 98
B. 98√3
C. 125√3
D. 196√3
The sides opposite the
[30^\circ],
[60^\circ], and right angles of every
[30^\circ]-
[60^\circ]-
[90^\circ] triangle relate in a
[1 : √3 : 2] ratio respectively.
2 / √3 = 196 / H
H = 196 (√3 / 2)
H = 98√3
A washing machine is being redesigned to handle a greater volume of water. One part is a pipe with a radius of
[3] centimeters and a length of
[11cm}]. It gets replaced with a pipe of radius
[4cm}], and the same length. The new pipe can hold Wπ more cubic centimeters of water than the old pipe, where [w] is a constant. What is the value of [w]?
The initial base of the pipe is a circle of radius 3 cm. So the area of the base is= π(3)^2 * 11 = 99π cm^3
Initial volume= 9π * 11 = 99π cm^3
The final volume = π(4)^2 * 11 = 176 cm^3
176π - 99π = 77π
The value of w is 77
A confectioner is applying a sugar coating to some spherical candies. The given equation approximates
[V], the volume of coating in cubic millimeters, needed to create a
[t]-millimeter thick coating on a candy with a surface area of
[A] square millimeters. If
[20\pi] cubic millimeters of coating is needed to create a
[0.2]-millimeter thick coat on a spherical candy, what is the uncoated candy’s radius in millimeters?
V = At
Surface area of a sphere: A = 4πr^2
V = At
V = (4πr^2)t
20π = (4πr^2)(0.2)
20π = 0.8πr^2
25 = r^2
r = +- 5
r = 5
3.5(x + 2.2)^2 + 3.5(y - 11.1)^2 - 21 = 0
What is the radius of the circle to the nearest hundredth?
3.5(x + 2.2)^2 + 3.5(y - 11.1)^2 - 21 = 0
(x + 2.2)^2 + (y - 11.1)^2 = 21 / 3.5
(x + 2.2)^2 + (y - 11.1)^2 = 6
√6 = 2.45
The sector of a circle shown has center at [O]. The radius of the circle has length [5]. The arc [XZY] has length [7]. To the nearest tenth, what is the shaded area of the sector?
A = 1/2 O * r^2 — area of a circular sector
S = rO — length of a circular arc
S = rO
7 = 5 * O
7 / 5 = O
1.4 = O
A = 1/2 O * r^2
= 1/2 (1.4) * 5^2
= (0.7) * 25
= 17.5
A circle in the
[xy]-plane has the equation 36x^2 + 36y^2 - 12x - 27y - 8 = 0.
How long is the radius of the circle?
A. 5 / 8
B. 25 / 64
C. 2 / 9
D. √2 / 3
(36x^2 - 12x) + (36y^2 - 27y) = 8
Now, we can divide the entire equation by 36:
( x^2 - 1/3x) + (y^2 - 2/9) = 2/9
( x^2 - 1/3x + 1/36) + (y^2 - 3/4y + 9/64) = 2/9 + 1/36 + 9/54
= 25/64
r^2 = 25/64
r = +- 5/8
