Question 1 774 men from the New Model Army, 226 from the New Model Army, and 226 from the Royalist Army were included in a random sample of the 774 soldiers who took part in the Battle of Preston. Use a 0.05 significance level to test the claim that fewer than one quarter of the soldiers were Royalist. Answer 1 Step 1 Given: 𝑋 = 226 , 𝑛 = 774 X=226,n=774 Sample proportion: 𝑝 = 226 774 ≈ 0.292 p= 774 226 ≈0.292 Null and alternative hypotheses: 𝐻 0 : 𝑝 = 0.25 , 𝐻 1 : 𝑝 < 0.25 H 0 :p=0.25,H 1 :p<0.25 Step 2 The standardized test statistic: 𝑍 = 𝑝 ^ − 𝑝 0 𝑝 0 ( 1 − 𝑝 0 ) 𝑛 = 0.292 − 0.25 0.25 ( 1 − 0.25 ) 774 ≈ 2.698 Z= n p 0 (1−p 0 ) p ^ −p 0 = 774 0.25(1−0.25) 0.292−0.25 ≈2.698 Step 3 Critical value at 𝛼 = 0.05 α=0.05: 𝑍 𝑐 = − 1.645 Z c =−1.645 Conclusion: Since 𝑍 = 2.698 > − 1.645 , we reject the null hypothesis. Since Z=2.698>−1.645, we reject the null hypothesis. Therefore, fewer than one quarter of the soldiers were Royalist. Question 2 Find the domain of the vector-valued function: 𝑟 ( 𝑡 ) = 9 − 𝑡 2 𝑖 + 𝑡 2 𝑗 − 5 𝑡 𝑘 r(t)= 9−t 2 i+t 2 j−5tk Answer 1 Step 1 Given: 𝑟 ( 𝑡 ) = 9 − 𝑡 2 𝑖 + 𝑡 2 𝑗 − 5 𝑡 𝑘 r(t)= 9−t 2 i+t 2 j−5tk To find the domain: The square root is defined when: 9 − 𝑡 2 ≥ 0 ⟹ − 3 ≤ 𝑡 ≤ 3 9−t 2 ≥0⟹−3≤t≤3 Answer: The domain is 𝑡 ∈ [ − 3 , 3 ] t∈[−3,3]. Question 3 Find the exact length of the curve. Use a graph to determine the parameter interval: 𝑟 = cos ⁡ 2 ( 𝜃 2 ) r=cos 2 ( 2 θ ) Answer 1 Using the polar arc length formula: 𝐿 = ∫ 𝑎 𝑏 𝑟 2 + ( 𝑑 𝑟 𝑑 𝜃 ) 2 𝑑 𝜃 L=∫ a b r 2 +( dθ dr ) 2 dθ Given 𝑟 = cos ⁡ 2 ( 𝜃 2 ) r=cos 2 ( 2 θ ), compute: 𝑑 𝑟 𝑑 𝜃 = cos ⁡ ( 𝜃 2 ) ( − sin ⁡ ( 𝜃 2 ) ) = − 1 2 sin ⁡ ( 𝜃 ) dθ dr =cos( 2 θ )(−sin( 2 θ ))=− 2 1 sin(θ) Substitute into arc length formula: 𝐿 = ∫ 0 2 𝜋 cos ⁡ 4 ( 𝜃 2 ) + ( − 1 2 sin ⁡ ( 𝜃 ) ) 2 𝑑 𝜃 L=∫ 0 2π cos 4 ( 2 θ )+(− 2 1 sin(θ)) 2 dθ Using symmetry of the curve, simplify and integrate over [ 0 , 𝜋 ] [0,π]: 𝐿 = 4 L=4 Question 4 Change from rectangular to spherical coordinates ( 𝜌 ≥ 0 , 0 ≤ 𝜃 ≤ 2 𝜋 , 0 ≤ 𝜙 ≤ 𝜋 ρ≥0,0≤θ≤2π,0≤ϕ≤π): a) ( 0 , − 8 , 0 ) (0,−8,0) b) ( − 1 , 1 , − 2 ) (−1,1,−2) Answer 1 Part a) Rectangular: ( 0 , − 8 , 0 ) (0,−8,0). 𝜌 = 𝑥 2 + 𝑦 2 + 𝑧 2 = 0 2 + ( − 8 ) 2 + 0 2 = 8 ρ= x 2 +y 2 +z 2 = 0 2 +(−8) 2 +0 2 =8 𝜙 = arccos ⁡ ( 𝑧 𝜌 ) = arccos ⁡ ( 0 8 ) = 𝜋 2 ϕ=arccos( ρ z )=arccos( 8 0 )= 2 π 𝜃 = arctan ⁡ ( 𝑦 𝑥 ) = arctan ⁡ ( − 8 0 ) = 3 𝜋 2 θ=arctan( x y )=arctan( 0 −8 )= 2 3π Spherical: ( 8 , 3 𝜋 2 , 𝜋 2 ) (8, 2 3π , 2 π ). Part b) Rectangular: ( − 1 , 1 , − 2 ) (−1,1,−2). 𝜌 = 𝑥 2 + 𝑦 2 + 𝑧 2 = ( − 1 ) 2 + 1 2 + ( − 2 ) 2 = 6 ρ= x 2 +y 2 +z 2 = (−1) 2 +1 2 +(−2) 2 = 6 𝜙 = arccos ⁡ ( 𝑧 𝜌 ) = arccos ⁡ ( − 2 6 ) ϕ=arccos( ρ z )=arccos( 6 −2 ) 𝜃 = arctan ⁡ ( 𝑦 𝑥 ) = arctan ⁡ ( 1 − 1 ) = 3 𝜋 4 θ=arctan( x y )=arctan( −1 1 )= 4 3π Spherical: ( 6 , 3 𝜋 4 , arccos ⁡ ( − 2 6 ) ) ( 6 , 4 3π ,arccos( 6 −2 )). Question 5 Transform the polar function 𝑟 = − 4 sin ⁡ ( 𝜃 ) r=−4sin(θ) to rectangular coordinates and graph. Answer 1 Given: 𝑟 = − 4 sin ⁡ ( 𝜃 ) r=−4sin(θ). 𝑟 2 = − 4 𝑟 sin ⁡ ( 𝜃 ) ⟹ 𝑥 2 + 𝑦 2 = − 4 𝑦 r 2 =−4rsin(θ)⟹x 2 +y 2 =−4y 𝑥 2 + 𝑦 2 + 4 𝑦 = 0 ⟹ 𝑥 2 + ( 𝑦 + 2 ) 2 = 4 x 2 +y 2 +4y=0⟹x 2 +(y+2) 2 =4 This represents a circle with center ( 0 , − 2 ) (0,−2) and radius 2 2.