University:
California State University, Northridge
Course:
MATH 262 | Introduction to Linear Algebra
Academic year:

2024
Views:

240

Pages:

4
Author:

Beckett Gray

Question 1 Define and explain the perimeter of a circle. Answer 1 Step 1: Perimeter is the distance around a closed figure. For a circle, the perimeter is called the circumference. Formulas for Circumference: Given the radius 𝑅 R: 𝐶 = 2 𝜋 𝑅 . C=2πR. Given the diameter 𝐷 D: 𝐶 = 𝜋 𝐷 ( where 𝐷 = 2 𝑅 ) . C=πD(where D=2R). If the area 𝐴 A of the circle is known, the circumference can be found as: 𝐶 = 4 𝜋 𝐴 . C= 4πA . Step 2: Illustration of a circle with labeled radius 𝑅 R. Question 2 How does the sum of the first odd integers relate to the geometric model? Can even integers be represented similarly? Answer Step 1: The geometric model shows squares being built by adding layers of circles: 1 = 1 2 1=1 2 (1 circle). 1 + 3 = 4 = 2 2 1+3=4=2 2 (4 circles, forming a 2x2 square). 1 + 3 + 5 = 9 = 3 2 1+3+5=9=3 2 (9 circles, forming a 3x3 square). … …. Thus, the sum of the first 𝑛 n odd integers equals 𝑛 2 n 2 : 1 + 3 + 5 + ⋯ + ( 2 𝑛 − 1 ) = 𝑛 2 . 1+3+5+⋯+(2n−1)=n 2 . Step 2: For even integers: Adding even integers 2 , 4 , 6 , … 2,4,6,… does not form squares. The last term is 𝑛 ( 𝑛 + 1 ) n(n+1), which is not represented in the geometric model. Question 3 A student runs around a track 7 times. The track consists of a rectangle and two semicircles. Find the total distance. Answer Step 1: Given: Diameter of semicircle 𝐷 = 7 m D=7m, length of rectangle 𝑙 = 15 m l=15m. Total distance in one round: 𝑑 = 𝜋 𝐷 + 2 𝑙 . d=πD+2l. Step 2: Substitute values: 𝑑 = 𝜋 ( 7 ) + 2 ( 15 ) . d=π(7)+2(15). Using 𝜋 = 22 7 π= 7 22 : 𝑑 = 22 7 ( 7 ) + 30 = 22 + 30 = 52 m . d= 7 22 (7)+30=22+30=52m. Step 3: Distance for 7 rounds: Total distance = 7 × 52 = 364 m . Total distance=7×52=364m. Final Answer: 364 m 364m. Question 4 Are two circles with the same center always congruent? Answer No, the given sentence is false. Concentric circles share the same center but can have different radii. Congruent circles must have the same radius. Thus, two circles can be concentric but not congruent. Question 5 If three concentric circles have radii in the ratio 2 : 3 : 5 2:3:5, what is the ratio of their arc lengths for an 8 0 ∘ 80 ∘ angle? Answer Step 1: Given radii ratio 2 : 3 : 5 2:3:5, and the arc lengths correspond to radii for the same central angle. Step 2: Since arcs are proportional to radii, the ratio of arc lengths is also: 2 : 3 : 5. 2:3:5. Final Answer: 2 : 3 : 5 2:3:5.