Question

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#1 A cone-shaped paperweight has a diameter of 3 inches and a height of 5 inches. What is the volume of the paperweight? Use 3.14 for pi.

#2 A rectangle has an area of 384 m². The length and the width of the rectangle are changed by a scale factor of 0.75. What is the area of the new rectangle?

#3 The sector of a circle with a 16-centimeter radius has a central angle measure of 45°. What is the exact area of the sector in terms of π ?

#4 A chair is placed near a 26-ft tall tree. A right triangle is formed between the chair, the base of the tree, and the top of the tree. The angle formed at the top of the tree is 56°. How far from the tree is the chair? Enter your answer as a decimal in the box. Round only your final answer to the nearest tenth.

#5 The equation for a circle is ​x2+10x+y2+12y+52=0​ .

What is the equation of the circle in standard form?

(x+25)2+(y+36)2=36
or
(x+25)2+(y+36)2=9
or
(x+5)2+(y+6)2=9
or
(x+5)2+(y+6)2=36

#6 What is the equation of a parabola with (−2, 4) as its focus and y = 6 as its directrix?

#7 Including the bus driver, there are 15 people on a bus. During the bus ride, each person produces 750 BTUs (British thermal units). The interior of the bus measures 38 feet by 8.5 feet by 6.25 feet.

How many BTUs per cubic foot are produced?

#8 A wooden cube of bamboo has a side length of 7 cm and a mass of 120 g.

What is the density of the cube? ____ g/cm^3

Answer 1

Question 1

Volume of cone =
`(1)/(3) \pi r^(2) h= (1)/(3) \pi (1.5)^(2) ( 5)=11.775`

Question 2

Scale factor of length = 0.75
Scale factor of area = 0.75² =0.5625
Area of original rectangle = 384
Area of the resized rectangle = 216

Question 3

Radius = 16
Angle = 45°
Area =
`(45)/(360)`×
`\pi (16)^(2)`=
`32 \pi`

Question 4
Using the trigonometry ratio

`tan(x)= (opposite)/(adjacent)`

`tan (56)= (opposite)/(26)`
opposite side =
`26tan(56)=39` ft to the nearest ten

Question 5
Circle equation is given by
`(x-a)^(2)+ (y-b)^(2)= r^(2)`

`x^(2) +10x+ y^(2)+12y+50=0`, using the completing the square

`(x+5)^(2)-25+ (y+6)^(2)-36+52=0`

`(x+5)^(2)+ (y+6)^(2)=-52+36+25`

`(x+5)^(2)+ (y+6)^(2)=9`

Question 6
First, we need to find the distance between
`( x_(0), y_(0) )` and the focus.
`( x_(0), y_(0) )` is any point on the parabola. We also need to find the distance between
`( x_(0), y_(0) )` and the directrix

The distance between
`( x_(0), y_(0) )` and (-2,4) is given

`\sqrt{( x_(0)+2) ^(2)+ ( y_(0)-4)^(2) }`

The distance between
`( x_(0), y_(0) )` and y=6 is given
`| y_(0)-6|`

Equating the two expressions gives

`\sqrt{( x_(0)+2) ^(2)+ ( y_(0)-4)^(2) }` =
`| y_(0)-6|`
Squaring both sides give

`(x_(0)+2) ^(2)+ ( y_(0)-4)^(2)= (y_(0)-6)^(2)`
Expand and simplify

`x_(0) ^(2)+4 x_(0) +4 y_(0)-16=0`

`y_(0)= (x_(0) )/(4)- x_(0)+16`

Question 7

Total BTU of 15 people on the bus = 15×750 = 11250 BTU
The volume of the bus = 38×8.5×6.25=2018.75 cubic foot
BTU per cubic foot = 11250 ÷ 2018.75 = 5.57

Question 8

Density = Mass/Volume = 120÷(7³) = 0.35 (two decimal places)

Diagram 1
HIJF is a quadrilateral. Angle IHF and IJF makes right-angle to the radius. It leaves us with angle HJF remain unknown.
Angle HJF = 180°-101° = 79°
Length of arc HJ =
`(79)/(360)*281=61.7` (round 1 dp)

Diagram 2
By using the trigonometry ratio

`sin(x)= (opposite)/(hypotenuse)`

`sin(x)= (15)/(115)`

`sin^(1)( (15)/(115))=x`

`x=7.5`

Diagram 3
Area of triangle =
`(1)/(2)(8)(13)sin(33)=28.32`

Diagram 4
Let O be the centre of the circle
Angle AOC = 25×2=50°