University:
Carleton College
Course:
MATH 120 | Calculus 2
Academic year:

2024
Views:

319

Pages:

5
Author:

Nevaeh Bauer

Question 1 What is the volume of this triangular pyramid? What is the volume of this cylinder? Use π ≈ 3.14 and round your answer to the nearest hundredth. Answer Step 1 In this question we will use basic of geometry i.e formula of volume of triangular pyramid and volume of cylinder. We will use volume of triangular pyramid = 1/3 * Area of base *Height Volume of cylinder = πr²h Solution of the above problem is in the step below. (a) As we know the formula of Volume of triangular pyramid So, Volume of pyramid = 1/3 × Area of base × Height As we can observe that Area of base = 1/2 × B × h = 1/2 × 15 × 10 = 75 m² Height of triangular pyramid =8m Hence, Volume of pyramid = 1/3 × 75 × 8 = 200 m³ Volume of triangular pyramid = 200 m³ (b) As we know that the volume of cylinder = πr²h Where; r= radius of cylinder =1.8 ft h= Height of cylinder =4 ft So, Volume of cylinder = π(1.8)²(4) Volume of cylinder = 40.6944 ft³ Question 2 Use a graphing utility to graph the given function and the equations y = |x| and y = -|x| in the same viewing window. Using the graphs to observe the Squeeze Theorem visually, find lim f(x). f(x) = x cos(1/x) Answer 1 Step 1 Given function is h(x) = x cos(1/x) we have to find lim f(x) x->0 Step 2 We know that cosine function is always between -1 and 1 and is given function is always between -|x| and |x| which both go to zero as x ⇒ 0. h(x) = xcos(1/x), y1 = |x| and y2 = -|x| Since -1 ≤ cos (1/x) ≤ 1 for all x ≠ 0 it follows that y2 ≤ h(x) ≤ y1 for all x ≠ 0 But lim y2 = lim y1 = 0 x->0 x->0 Therefore, the squeeze theorem can be used to conclude that lim f(x) = lim x cos(1/x) = 0 x->0 x->0 Answer 2 Step 1 Given: f(x) = x cos(1/x) Use a graphing utility to graph the given function and the equations y = |x| and y = -|x| Step 2 Explanation: The lower and upper functions have the same limit at x = 0. The middle function has the same limit value because it is trapped between the two outer functions. Step 3 Squeeze Theorem: Suppose f(x) ≤ g(x) ≤ h(x) for all x in an open interval about "a". lim f(x) = lim h(x) = L x->a x->a Then, lim g(x) = L x->a At x = 0, lim[-|x|] = lim[|x|] = 0 x->0 x->0 Then, lim x cos(1/x) = 0 x->0 Answer 3 lim f(x) = lim |x| = 0 x->0 x->0 = 0.1 lim f(x) = 0 x->0 Question 3 Explain, please, what is the measure of one interior angle of a regular nonagon? Answer All sides are the same length and all interior angles are the same size. To find the measure of the angles, we know that the sum of all the angles is 1260 degrees (from above). And there are nine angles. 1260 ÷ 9 = 140 So, the measure of the angle of a regular nonagon is 140 degrees. Answer 2 Sum of all interior angles of any convex N-sided polygon equals to (N-2)180° In N-sided regular polygon all N angles are equal, so each is equal to ((N-2)/N) × 180° ((9-2)/9) × 180° = 140° Answer 3 Each interior angles of a regular nonagon measures 140 degree. Question 4 Using analytic geometry, find the coordinates of the point on the x axis which is equidistant from A(5, 8) and B(-3, 4) Answer Step 1 Given, A(5, 8), B(-3, 4) Let, the equidistant point be P(x, 0) So, using distance formula: √(x-5)² + (0-8)² = √(x + 3)² + (0-4)² Both Sides: (x-5)² + (0-8)² = (x + 3)² + (0-4)² x² + 25 - 10x + 64 = x² + 9 + 6x + 16 25 + 64 - 9 - 16 = 10x + 6x 16x = 64 x=4 Coordinate is (4, 0)