Precal unit 10 ANSWERS

Find the polar representation of the point: (1,π3) The terminal side of the angle rotates 2π counterclockwise.

(1,7π/3)

Convert to polar coordinates for r and −r to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (√3,1.26) _____ _____

(2.14,π/5) (−2.14,6π/5)

Convert to rectangular coordinates (x,y) . If necessary, round to the nearest hundredth. (r,θ)=(5,π/4)

(3.54,3.54)

Solve. Gravitation acceleration is 32 ft/sec. Darnell kicked a football from the ground at an angle of 45° from horizontal. The initial velocity of the football was 120 feet/second. At 2 seconds into the flight, how high was the ball ( y=______) and how far downfield was it ( x=_____)?

169.70 feet x=(vicosθ)t+xi x=(120cos45∘)2+0 x=(120(0.7071))2+0≈169.70 y=−12at2+(visinθ)t+yi y=−12(32)22+(120sin45∘)2+0 y=−64+(120(0.7071))2+0≈105.70

Choose the word from the list that names the type of graph and write it on the blank.

Cardioid A Cardioid is a special Limaçon where a=b.a=b. It is in the shape of a heart.

Choose the word from the list that names the type of graph and write it on the blank.

Lemniscate

Choose the word from the list that names the type of graph and write it on the blank.

Lemniscate A Lemniscate is in the shape of a figure 8 running diagonally.

Choose the word from the list that names the type of graph and write it on the blank.

Limaçon A Limaçon has equation r=a±bsinθ or r=a±bcosθ

pole

a fixed point called the origin in a polar coordinate system

composed

made up of a combination of parts

orientation

position or location relative to the previous point(s)

eliminate

remove

Find the table values to the nearest hundredth, if needed, using a calculator, if necessary. x=t+2x=t+2 and y=t2,y=t2, (−2≤t≤2)(-2≤t≤2) t -2 −1 0 1 2 x y

t -2 -1 0 1 2 x 0 1 2 3 4 y 4 1 0 1 4

The parameter, t, is often used to represent _____.

time

Eliminate the parameter: x = 7^5t and y = 7^5t − 2

x = 49y

Eliminate the parameter: x=4t^2+2 and y=t−2

x=4y^2+16y+18

Eliminate the parameter: x = cosθ and y = 5sinθ

x^2 + y^2/25 = 1

Solve. Gravitation acceleration is 32 ft/sec. Ron kicked a soccer ball off the ground at an angle of θ. The initial speed of the ball was 50 feet per second. Write the two simplified parametric equations for the path of the ball.

y=−162+(50sinθ)t x=(50cosθ)t

Convert to rectangular coordinates (x,y). If necessary, round to the nearest hundredth. (r,θ)=(−3,−2π/3)

(1.5,2.6)

Find the polar representation of the point: (5,25°), The terminal side of the angle rotates 360° counterclockwise.

(5,385°)

A polar point lies on the terminal side of an angle drawn from the polar axis. The coordinates of the point are (6, 55°). The side rotates counterclockwise 720 °. What is the new polar representation of the point?

(6, 775∘)

Solve. Point A lies 7 units from the pole on the terminal side of an angle π9 from the polar axis. Find the new coordinates if the side is rotated 2π counterclockwise.

(7,19π/9)

Convert to rectangular coordinates (x,y). If necessary, round to the nearest hundredth. (r,θ)=(−2,π3)

(−1,−1.73)

Convert to polar coordinates for (r,θ) and (−r,θ) to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (√3,1) _____ _____

(−2,7π6) (2,π6)

Find the polar representation of the point: (−3,5π4), The terminal side of the angle rotates 2π clockwise.

(−3,−3π/4)

Choose the word from the list that names the type of graph and write it on the blank.

Rose Curve A rose curve has "petals" that make it look like a flower.

parameter

an independent variable upon which two other variables depend

Choose the word from the list that names the type of graph and write it on the blank.

circle A Circle has equation r=asinθor r=acosθ

ordered pair

coordinates such as (2, 4) that you would place on a graph

Two variables depend upon the _____.

parameter

Solve. Gravitation acceleration is 32 ft/sec. Cale hit a "line drive" from 4 feet above and parallel to the ground. The ball left the bat at a velocity of 105 feet per second. how far did the ball travel before it hit the ground? Find to the nearest whole number.

x=53 feet

Eliminate the parameter: x=4cosθ and y=2sinθ

x^2/16+y^2/4=1

Eliminate the parameter: x=3√cosθ and y=5sinθ

x^2/3+y^2/25=1

Eliminate the parameter: x=2cosθ and y=6sinθ

x^2/4+y^2/36=1

Eliminate the parameter: x = 3cosθ and y = 3sinθ

x^2/9 + y^2/9 = 1

Eliminate the parameter: x = t − 5x and y = 2t + 1

y = 2x + 1 x = t − 5 so t=x+5. Substitute into the other parametric equation: y = 2(x + 5) + 1 = 2x + 11

Eliminate the parameter: x = t + 9 and y = t − 3

y = x − 12

Solve. Gravitation acceleration is 32 ft/sec. Julian shoots a flare from the deck of a ship 50 feet above the water. The initial velocity of the flare is 83 feet/second. If Julian shoots the flare straight up, how high will the flare be in 4 seconds?

y=126 feet

Eliminate the parameter: x=4^7t and y=4^7t+2

y=16x

Eliminate the parameter: x=t and y=2t+1

y=2x+1

Eliminate the parameter: x=√2t and y=t^2+1

y=x^2/2+1

Eliminate the parameter: x=t+3 and y=t−7

y=x−10

Convert to rectangular coordinates (x,y). If necessary, round to the nearest hundredth. (r,θ)=(−1,225°)

≈(0.7,0.7)

Convert to rectangular coordinates (x,y). If necessary, round to the nearest hundredth. (r,θ)=(−3,−3π/4)

(2.12,2.12)

Find the polar representation of the point: (−4,3π/4) , The terminal side of the angle rotates π clockwise.

(−4,−π/4)

Finding a rectangular equation from parametric equations is called eliminating the _______________.

parametric

The equations containing the parameter are called _____ equations.

parametric

Convert to polar coordinates for (r,θ) and (-r,θ) to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (1,1) _____ _____

(2√,π4) (−2√,5π4)

rotation

moving around a fixed point

Convert to polar coordinates for r and −r to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (1.54,0.5) _____ _____

(1.62,π/10) (−1.62,11π/10)

Convert to rectangular coordinates (x,y). If necessary, round to the nearest hundredth. (r,θ)=(4,π/3)

(2,3.5)

Convert to polar coordinates for (r,θ) and (-r,θ) to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (3√,−1) _____ _____

(2,−π/3) (−2,2π/3)

Solve.Find the polar representation of a point located 2 units from the pole on the terminal side of an angle 3π4 from the polar axis after the side rotates 3π clockwise and r=−2.

(−2,−9π/4)

Convert to polar coordinates for (r,θ) and (-r,θ) to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (1.5,2.06) _____ _____

(−2.5,13π/10) (2.5,3π/10)

Convert to polar coordinates for (r,θ) and (-r,θ) to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (3,3) _____ _____

(−3√2,5π/4) (3√2,π/4)

Convert to polar coordinates for (r,θ) and (−r,θ) to the nearest hundredth. Use the Tangent Table. Use the closest approximation to a ππ answer in column 1 or 8 for the angle. (3√,5.33) _____ _____

(−5.6,7π/5) (5.6,2π/5)

interval

a set of real numbers including the first and last numbers and any real number between them within the set

Two equations containing the same parameter are called _____ equations.

parametric

Find the table values to the nearest hundredth, if needed, using a calculator, if necessary. x = t^2 and y = t + 1, (−2 ≤ t ≤ 2) t −2 -1 0 1 2 x __|__|__|__|__| y __|__|__|__|__|

t −2 -1 0 1 2 x _-4_|_-1_|_0_|_1_|_4_| y _-1_|_0_|_1_|_2_|_3_|

Eliminate the parameter: x = t^2 − 4 and y = t + 3

x = y^2 − 6y + 5

Finding a rectangular equation from parametric equations is called eliminating the _______________.

parameter

Two equations containing the _____ are called parametric equations.

parameter

Find the table values to the nearest hundredth, if needed, using a calculator, if necessary. x=t+3x=t+3 and y=1−3t,y=1-3t, (−2≤t≤2)(-2≤t≤2) t −2 −1 0 1 2 x y

t −2 −1 0 1 2 x _1_ _2_ _3_ _4_ _5_ y _7_ _1_ _4_ _-2_ _-5_

Find the table values to the nearest hundredth, if needed, using a calculator, if necessary. x = t − 1x = t - 1 and y = 1 − t, (−2 ≤ t ≤ 2)y = 1 - t, (-2 ≤ t ≤2) t −2-2 −1-1 00 11 22 x _____ ____ ____ ____ ____ y _____ _____ ____ ____ ____

t −2-2 −1-1 00 11 22 x __-3___ __-2__ __-1__ __0__ __1__ y __3___ ___2__ __1__ __0__ __-1__

Eliminate the parameter: x = 2t^2 − 2 and y = 2t + 2

x = y^2/2 − 2y

Solve. Gravitation acceleration is 32 ft/sec. Ian hit a baseball from 2 feet off the ground at an angle of θ.θ. It left the bat at a speed of 100 feet per second. Write the simplified parametric equations for the path of the ball. x= y=

x= (100cosθ)t y=−16t2+(100sinθ)t+2

Solve. Gravitation acceleration is 32 ft/sec. Julian shoots a flare from the deck of a ship 50 feet above the water. The initial velocity of the flare is 83 feet/second. Write the set of parametric equations for the path of the flare.

x=(83cosθ)t y=−16t^2+(83sinθ)t+50

Solve. Gravitation acceleration is 32 ft/sec. Abigail kicked a soccer ball from the ground at an angle of θ . The initial velocity of the ball was 100 feet/second. Write the simplified parametric equations for the path of the ball.

x=100cosθt y=−16^2+(100sinθ)t

Solve. Gravitation acceleration is 32 ft/sec. Katie fired an air gun 4 feet from and parallel to the ground. The velocity of the bullet leaving the gun was 385 fps (feet per second). How far did the bullet travel before hitting the ground?

x=192.5 feet

Solve. Gravitation acceleration is 32 ft/sec. Elisha hit a tennis ball 4 feet above and parallel to the ground. The ball left her racquet at a velocity of 120 feet per second. To the nearest foot, how far did the ball travel before it hit the ground?

y=−16^2+(120sinθ)t

Solve. Gravitation acceleration is 32 ft/sec. Nia threw a softball at an initial velocity of 72 feet per second (fps) and at an angle of elevation of θ degrees. The ball left her hand 3 feet above the ground. Write the two parametric equations for the path of the ball.

y=−16t^2+(72sinθ)t+3 x=(72cosθ)t

Find the table values to the nearest hundredth, if needed, using a calculator, if necessary. x=cosθ and y=sinθ, (0∘≤θ≤180∘) θ 0∘ 15∘ 30∘ 45∘ 60∘ 75∘ 90∘ x __ __ __ __ __ __ y __ __ __ __ __ __

θ 0∘ 15∘ 30∘ 45∘ 60∘ 75∘ 90∘ x _1.00_|_0.97_|_0.87_|_0.71_|_0.5_|_0.26_|_0.00_ y _0.00_|_0.26_|_0.5_|_0.71_|_0.87_|_0.97_|_1.00_

Find the table values to the nearest hundredth, if needed, using a calculator, if necessary. x=4cosθx=4cosθ and y=2sinθ, (0∘≤θ≤180∘)y=2sinθ, (0∘≤θ≤180∘) θ 0∘ 30∘ 60∘ 90∘ 180∘ x y

θ 0∘ 30∘ 60∘ 90∘ 180∘ x _4_ _3.5_ _2_ _0_ _-4_ y _0_ _1_ _1.7_ _2_ _0_

Find the table values to the nearest hundredth, if needed, using a calculator, if necessary. x=2+3cosθx=2+3cosθ and y=1+sinθy=1+sinθ θ 0∘ 30∘ 60∘ 90∘ 180∘ 270∘ x y

θ 0∘ 30∘ 60∘ 90∘ 180∘ 270∘ x 5 4.60 3.50 2 -1 2 y 1 1.5 1.87 2 1 0