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

2024
Views:

294

Pages:

4
Author:

Beckett Gray

Question 1 The diameter of a U.S quarter is 24.26 millimeters. How long is a line of 37 quarters laid edge to edge? Answer Step 1: Diameter of one U.S quarter = 24.26 mm =24.26mm. Total distance = number of quarters × diameter =number of quarters×diameter. Length of a line of 37 quarters = 37 × 24.26 = 897.62 mm . Length of a line of 37 quarters=37×24.26=897.62mm. Question 2 Inside the quarter circle are two semicircles. Which has a greater area, A or B? Answer Step 1: The quarter circle has radius 𝑟 r, and the semicircles inside have radii of 𝑟 2 2 r . Area of one semicircle: Area = 𝜋 ( 𝑟 2 ) 2 2 = 𝜋 𝑟 2 8 . Area= 2 π( 2 r ) 2 = 8 πr 2 . Area of the quarter circle: Area = 𝜋 𝑟 2 4 . Area= 4 πr 2 . Step 2: Sum of the two semicircles' areas: 2 × 𝜋 𝑟 2 8 = 𝜋 𝑟 2 4 . 2× 8 πr 2 = 4 πr 2 . The area of the quarter circle is exactly equal to the sum of the areas of the two semicircles. Conclusion: Both areas 𝐴 A and 𝐵 B are equal. Question 3 Find the area of the shaded region. Answer Step 1: Length of the rectangle = 18 cm 18cm. The two circles have equal diameters, and their sum equals the rectangle's length: 2 𝑥 = 18 ⟹ 𝑥 = 9 cm . 2x=18⟹x=9cm. Radius of one circle: 𝑟 = 𝑥 2 = 9 2 = 4.5 cm . r= 2 x = 2 9 =4.5cm. Step 2: Area of the rectangle: 𝐴 rectangle = length × breadth = 18 × 9 = 162 cm 2 . A rectangle =length×breadth=18×9=162cm 2 . Area of one circle: 𝐴 circle = 𝜋 𝑟 2 = 𝜋 ( 4.5 ) 2 ≈ 63.62 cm 2 . A circle =πr 2 =π(4.5) 2 ≈63.62cm 2 . Total area of two circles: 𝐴 circles = 2 × 63.62 = 127.24 cm 2 . A circles =2×63.62=127.24cm 2 . Step 3: Area of the shaded region: 𝐴 shaded = 𝐴 rectangle − 𝐴 circles = 162 − 127.24 = 34.76 cm 2 . A shaded =A rectangle −A circles =162−127.24=34.76cm 2 . Question 4 Find the area of the shaded region (the circles are edge to edge with all sides of the rectangle). Answer Length of the rectangle = 21 cm 21cm, breadth = 7 cm 7cm. Area of the rectangle: 𝐴 rectangle = 21 × 7 = 147 cm 2 . A rectangle =21×7=147cm 2 . Radius of each circle: 𝑟 = 7 2 = 3.5 cm . r= 2 7 =3.5cm. Area of one circle: 𝐴 circle = 𝜋 𝑟 2 = 𝜋 ( 3.5 ) 2 = 38.48 cm 2 . A circle =πr 2 =π(3.5) 2 =38.48cm 2 . Total area of three circles: 𝐴 circles = 3 × 38.48 = 115.45 cm 2 . A circles =3×38.48=115.45cm 2 . Area of the shaded region: 𝐴 shaded = 𝐴 rectangle − 𝐴 circles = 147 − 115.45 = 31.6 cm 2 . A shaded =A rectangle −A circles =147−115.45=31.6cm 2 . Correct Answer: Option A (31.6 cm²). Вы сказали: