Vector Calculus Questions and Answers

A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 96 feet. The height of the arch at a distance of 15 feet from the center is to be 9 feet. What will be the height of the arch at its center? Give your answer to 2 decimal places. 4 feet

Parametric equations of a line are x = 1 + 3t and y = 5 - 2t, teR. a. Write the coordinates of three points on this line. b. Show that the point P(-14, 15) lies on the given line by determining the parameter value of t corresponding to this point.

Based on FAA estimates the average age of the fleets of the 10 largest U.S. commercial passenger carriers is 13.4 years with a standard deviation of 1.7 years. Suppose that 40 airplanes were randomly selected from the fleets of these 10 carriers and were inspected for cracks in these airplanes that are considered too large for flying. What is the probability that the average age of these 40 airplanes is at least 14 years old? Round your answer to 4 decimal places. Remember to round your z-value to two decimal places. 0.0129 0.2654 0.3589 0.4731 None of the above

2s, The parametric equations of a line are given as x = -10 y = 8 +s, SER. This line crosses the x-axis at the point with coordinates A(a,0) and crosses the y-axis at the point with coordinates B(0, b). If O represents the origin, determine the area of the triangle AOB. -

The vectors ū1 = ū₂ = form a basis for R4. " ūz = ūs = Find the unique representation of any vector combination of u₁, U2, U3, and 4. O x = x₁ū₁ + (x₁1 + x2)µ2 + (X1 + X2 + X3)¯3 + (X1 + X2 + X3 + X4)ū4 ○ x = x₁ū₁ + (x1 − x₂)ū2 + (x₂ − X3)¯3 + (x3 − x4)ūṣ - = (x1, x2, x3, x4) E R4 as a linear ○ X = X₁Ū1 + (X1 + X2)ū2 + (x2 + X3)Ū3 + (x3 + x4) ū4 Ox₁u1 + (x₂ - ₁)₂ + (x3 - x₂)3 + (x₁ - x3) 4 = O x = x₁u₁ + x2Ū2 + x3Ū3 + x4¯¹

A ferris wheel with a radius of 20 feet has its center 28 feet above the ground. When in operation, it makes a full turn every 25 seconds. Let H(t) be the height of a point P on the ferris wheel t seconds after it begins operation, and suppose you know that P reaches the higihest point at t = 33 seconds. The maximum height is The first time P was at this height was a t = The next time P will be at this height will be at t = ofeet. We can model H(t) using a cosine function with amplitude A = phase shift 0 seconds. seconds. O and (enter the least possible value of the phase shift)

What is the component form of the vector represented by 8a when a = (-5,3), Enter the answer in the boxes. 8a-(0)

Let Q represent a mass of radioactive radium (226Ra) (in grams), whose half-life is 1599 years. The quantity of radium present after t years is 17/1599 Q = 5( ²12 ) Determine the quantity present after 300 years. Round to the nearest hundredth of a gram. ○ 4.39 g O 0g O 0.88 g O 0.47 g O 5.00 g

Question You have a coupon for a local craft store that is for 40% off one item. You would like to purchase a glue gun for $8.99 and a basket for $25.99. The coupon will take the discount of the highest-priced item. There will be a 6% sales tax added to the price of all items. What is the final price including all discounts and taxes? Report your answer to two decimal places. Do not include the dollar sign, $, in the answer box below..

What is the exact value of a trigonometric function of a given angle? Select True or False for each statement. Statement 3 2 cot (-1) = -√3 3 sin (27) = -1 cos (-5) = -- 3 True O O False O O O

Consider ABC, where a = 4, c= 8, and What are the measures of the missing angles and missing side? Enter your answer by filling in the boxes. Enter all values rounded to the nearest tenth. m/A- m/B- b= = m/C= 74°

Wyatt invested $1,600 in an account paying an interest rate of 5% compounded annually. Joshua invested $1,600 in an account paying an interest rate of 4% compounded quarterly. After 17 years, how much more money would Wyatt have in his account than Joshua, to the nearest dollar?

Using the standard Euclidean inner product in R² find an orthogonal basis for Wspan({(1, -1, 0, 1), (1, 0, 1, 1), (1, 2, 1, 3)}) List the vectors of this orthogonal basis by entering 1 vector into each box. If you have entered all your basis vectors, type "X" into any remaining answer boxes.

Find a polynomial f (x) of degree 5 that has the following zeros. -2, 4, 1, 6, -9 Leave your answer in factored form.

At 12:00pm, ship A is 124 kilometers west of ship B. Ship A is sailing north at 37 kilometers per hour whereas ship B is sailing south at 45 kilometers per hour. Determine the rate of change of the distance between the two ships at 7:00am. Round the solution to the nearest ten-thousandth, if necessary. ___- kilometers per hour

Perform the following mathematical operation, and report the answer to the correct number of significant figures. 0.97584 5.68 = [?]

What is the density of the object, reported to the correct number of significant figures? 16.400 g 9.1 mL = [?] L mL

Divide. (3x²+ 13x+11) (x+3) Your answer should give the quotient and the remainder.

How many significant figures should be recorded for the answer to the multiplication problem below? 24.32 x 0.010

An airplane flies at 500 mph with a direction of 135° relative to the air. The plane experiences a wind that blows 60 mph with a direction of 60°. Part A: Write each of the vectors in linear form. Show all necessary calculations. (6 points) Part B: Find the sum of the vectors. Show all necessary calculations. (2 points)

Vectors t = -61 – 2j, u = −4i – 8j, and v = -4i + 12j are given. Part A: Find the angle between vectors t and u. Show all necessary calculations.

Vector u PQ has initial point P (3, 17) and terminal point Q (9, 2). Vector v = RS has initial point R (30, 6) and terminal point S (8, 18). Part A: Write u and v in linear form. Show all necessary work. (4 points) Part B: Write u and v in trigonometric form. Show all necessary work. (8 points) Part C: Find 7u - 4v. Show all necessary calculations. (3 points)

Vectors t = -6i - 2j, u = -4i - 8j, and v = −4i + 12j are given. Part A: Find the angle between vectors t and u. Show all necessary calculations. (5 points) Part B: Choose a value for c, such that c> 1. Find w = cv. Show all necessary work. (2 points)

Calculate the area of the triangle with vertices: A (4, 2, -3), B (-1, -3, -1) and C (2, 2, 4)

Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) f(x)=sin(x) - 4 0<x< 2x

Solving linear equations Collect the variable terms on one side. Isolate the variable ➤ Check in the original equation. #2 Solve each equation for x. a) 2x - 5= 7-3(x + 1) Example: Solve 2 - 3(t - 2) = 5t Removing the parentheses: Combine like terms: Collect all variable terms on one side: Isolate the variable: Simplify Check: b) 2x + y = 3 2-3t+ 6 = 5t -3t+8 = 5t -3t-5t = -8 -8t = -8 -8t -8 -8 ⇒ t = 1 2-3(1-2) = 5(1) 2-3(-1) = 5 -8 2 + 3 = 5 true c) x-1= 5

Find the vector, not with determinants, but by using properties of cross products. (ix j) x k

Find the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1).

Simplify. sec²x-1/ tan²x

Which of the following IS NOT a form of the Pythagorean Identity? A. sin²x + cos²x = 1 B. 1 + cot²x = CsC²x C. tan³x + 1 = sec2x D. 1+tan²x = sin²x

Find two unit vectors orthogonal to both (9, 8, 1) and (-1, 1, 0). (smaller i-value)____ (larger i-value) ___

Find the cross product a × b. a = (2, 5, 0), b = (1, 0, 7)_____ Verify that it is orthogonal to both a and b. (a x b) a =___ (a x b) b =_____

Find a horizontal asymptote of f(x)= 3x² - 4x + 1 /x²+x-2

Round the number to 3 significant figures. 0.259 | 0100 Step 2: Now we need to decide if we round up or down. What is the answer rounded to three significant figures?

Which sentence below uses an incorrect homophone? A. Our assignment is to write a personal essay on the most memorable event of this past year. B. I can't seem to remember anything unless I right it down on paper. C. It was hard to accept, but we finally had to admit that she was absolutely right. D. I find that the best way to take notes it to write them down.

Which sentence uses the underlined word incorrectly? A. I only get minimal exercise in a day, even though I should get more. B. After a long time of questionable choices, I finally started to make mature decisions. C. I expect you to inspect your notes before you take your test tomorrow. D. I like to take a minimal pill everyday to make sure I am healthy. Academic Vocabulary X inspect - (v.) to examine, to study, to scrutinize E.g.: The manager would like to inspect our work before we move on to the next step. minimal - (adj.) least or smallest amount, negligible E.g.: We were disappointed at the minimal effort the team made to win the game. mature (adj.) grown-up, responsible, stable, established, settled, aged, ripe

Problem 2. Draw the geometric vectors v (¹) addition of geometric vectors in the plane. (2 marks in K/U, 1 mark in COM) and w= (9) on a plane. Add v and w using the definition of

Which sentence contains transitional words or phrases? A. As you can see, I am a better writer than a speaker. B. I am a better writer than a speaker. C. I am a good speaker. I am a better writer. D. I am a good speaker; I am a better writer.

If y = cos x, how many periods will there be between -5T and 9π? [?]

Round the number to 3 significant figures. 1.45 93 Step 2: Now we need to decide if we round up or down. What is the answer rounded to three significant figures? Enter

8cis(77). Evaluate the cube root of z when z = 8cis(-

Factor the polynomial completely. Enter the factored form without any spaces. Write any monomial factors first. 4a³ - 36a

A boat sails 285 miles south anc then 132 miles west. What is the magnitude of the boat's resultant vector? Hint: Draw a vector diagram. [?] miles

If y = -cos x, what x-value corresponds to a y-value of 1 between 0 and 2π?

Select all factors of the polynomial. 2y³ - 18y Multiple answers: Multiple answer Select one or more answers and submit. F To (y - 3) b (y-9) C (y + 3) d 2y di 2y²

A vector has a magnitude of 30 and a direction of 40°. Another vector has a magnitude of 70 and a direction of 130°. What are the magnitude and direction of the resultant vector? Round the magnitude to the thousandths place and the direction to the nearest degree.

Given u = -3i + 5j and v= -24i + 40j, are u and v parallel or orthogonal? Explain. O The vectors are orthogonal because cos 0 = 1. O The vectors are orthogonal because u v = 0. O The vectors are parallel because cos 0 = 1. O The vectors are parallel because u • v = 0.

66. A woman starts walking from home and walks 6 miles at 40° north of east, then 2 miles at 15° east of south, then 5 miles at 30° south of west. If she walked straight home, how far would she have to walk, and in what direction?

4. (28 points) Given the vector field F(x,y) = (ye* cos ex, sin e*), the clockwise oriented part of the x = t parabola C₁: y = -t² + 5t-6, the line segment C₂ from (3,0) to (2,0): te [2,3] Gify= a) Sketch C₁ and C₂

Which of the following represents vector u = -3i+ 8j in component form? Ou=(3,-8) Ou= (-3,8) Ou=(8,-3) Ou= (-8,3)