Vector Calculus Questions and Answers

If a varies jointly as b and c and a 27 when b 18 and c 3 find a when b 6 and c 1 3

Two children are playing with a ball The girl throws the ball to the boy The ball travels in the air curves 12 ft to the right and falls 37 ft away from the girl see the following figure If the plane th contains the trajectory of the ball is perpendicular to the ground find the plane s equation 37 ft X 12 ft

Differentiate g x log 37 3x 2 x 0

Find the derivative of the following function A 630 1 708 A

Consider the following curve 2 cos 20 25 Write an equation for the curve in terms of sin 8 and cos 8 Find a Cartesian equation for the curve Identify the curve ellipse circle Ohyperbola Oline Olima on Need Help Read

We are given the point in polar coordinates 9 7 We are asked to find two other pairs of polar coordinates to represent the point one with the radius r 0 and one with r 0 We can plot the point which creates an angle of counterclockwise with the polar axis in other words the positive horizontal axis The radius or distance from the origin is r 9 4 O 4 93 To find a second pair of polar coordinates for the same point that has radius r 0 we must use the same value r 9 This is the unique positive number to represent the distance of 9 from the point to the origin The angle as not unique however as adding any multiple of 2 results in the same angle Therefore we can add 2 to the angle of the given polar coordinates to find a second point as follows

Find a Cartesian equation for the curve 2 13 Identify the curve Oline O hyperbola O ellipse O lima on O circle

a 5 5 We are given the Cartesian coordinates 5 5 and must convert this point to polar coordinates Recall that to find polar coordinates r 8 we use the following equations 2 x y 5 tan 8 We can use these equations to find the following values for x y 5 5 2 x y tan 8 y X y X 5

Approximate to the nearest 0 1 degree all angles theta in the interval 0 degree 360 degree that satisfy the equation a sin theta 0 4530 b cos theta 0 9640 c tan theta 2 967 d cot theta 0 9405 e sec theta 1 419 f csc theta 1 588

1 2 Find the dominant eigen value of the following matrix Show all your work T 3

With the aid of a power reducing formula 2sin acos a Select the correct answer below O cos 4a O cos 2a cos 4a O cos 4a cos 2a cos 4a

Use a power reducing formula to simplify 16sin t Write your answer as an equivalent expression that does not contain any powers of sine or cosine greater than 1

For all values of a for which the expression is defined Select the correct answer below 2sin a 1 tan a 2tan a cos a sin a O2sin a tan a sec a 2tan a O tan 2a cos a

You are given that sin A 7 with A in Quadrant II and cos B 3 with B in Quadrant I Find sin A B Give your 17 answer as a fraction

You are given that sin A with A in Quadrant IV and sin B with B in Quadrant III Find cos A B Give 13 your answer as a fraction Provide your appr

7 Given the sequence an n 5 If the sequence converges indicate the 6 n value that it converges to If it diverges write diverges

Evaluate Sc xy x 2 ds along the curve r t 2ti tj 9 2t k 0 ts1 Sc xy x 2 ds Simplify your answer

32 e 4 t 3 At least one of the answers above is NOT correct 1 point The unique solution to the initial value problem Yo 6 is y t 8e 31 2 Determine the constant yo and the function g t help numbers g t 32e 41 3 I y y g t y 0 yo 32e 4 3 help formulas

Express the given quantity as a function f x of one variable x the perimeter of a rectangle of length x and width y that has an area of 187 square meters f x

Find the gradient field of the function f x y z 3x 3y 4z 1 2 The gradient field is Vf i j k

Given that the directional derivative of f at the point 1 5 5 in th direction of a 9i 6j 2k is 4 and that Vf 1 5 5 4 find Vf 1 5 5 f 1

1 point Suppose the clean water of a stream flows into Lake Alpha then into Lake Beta and then further downstream The in and out flow for each lake is 400 liters per hour Lake Alpha contains 500 thousand liters of water and Lake Beta contains 200 thousand liters of water A truck with 400 kilograms of Kool Aid drink mix crashes into Lake Alpha Assume that the water is being continually mixed perfectly by the stream a Let x be the amount of Kool Aid in kilograms in Lake Alpha hours after the crash Find a formula for the incremental change in the amount of Kool Aid Ax in terms of the amount of Kool Aid in the lake x and the incremental change in time Ar Enter At as Deltat Ax kg b Find a formula for the amount of Kool Aid in kilograms in Lake Alpha t hours after the crash kg x 1 c Let y be the amount of Kool Aid in kilograms in Lake Beta 1 hours after the crash Find a formula for the incremental change in the amount of Kool Aid Ay in terms of the amounts x y and the incremental change in time Ar Enter At as Deltat kg Ay d Find a formula for the amount of Kool Aid in Lake Beta 1 hours after the crash kg y t

Provide your answer below TT TT 2 2 34 1 0 1 2 TT 2 3 4 0 TT VI X J

1 Evaluate a sin sin 1 sin cos d b sin tan 2 3 e sin cos x c f cos tan tan sin 5 P Vp 9

1 Evaluate a sin sin 1 sin cos d 4 tanx secx 2 secx c 6 tanx secx 2 secx sin x cos x d sin x cos x 1 e sin x cos x secx sinx b 2 Simplify the following a sinx cos x secx cscx sin cos x b cos x sinx i sinxcosx 1 cos x sinx o tanx 1 COS X sin tan 2 cosx sinx 12 e sin cos x 3 c cos tan 3 tan sin 9 f

For matrices A B Mnxn F we say that A and B are unitarily similar if there exists a unitary matrix U E Mnxn F such that U AU B Note that since U is unitary U U so being unitarily similar is a special case of being similar Let A B Mnxn F be unitarily similar Use proof by induction to show that A and B are unitarily similar for all kEN

Given that tan x 2 what is tan x

Sketch the solid and a typical disk or washer y 4 2 4 y 4 2 4 4 06 4 2 4 y y 2 4

5 Find the number of sides for a polygon whose sum of the measures of its interior angles is a 900 b 1260

Given a generic point M in the Argand diagram representing the complex number z and let 8 iz 61 zi s t Re U 0 Determined the circle representing the locus of M Find the locus of M whenever the complex number U is real

graph be the two points that are labeled on the graph given below Notice that the sine unction is used in the answer template representing a sine function that is shifted and or reflected Use the variable x in your equation but be careful not use the multiplication x symbol 2 1 0 1 2 3 0 4 Provide your answer below TT 2 57 TT 3TT 2 3T 2 2TT

ind and sketch the level curves f x y c on the same set of coordinate axes for the given values of c Refer to these level curves as contour map z 3x y c 2 1 0 1 2 Choose the correct graph below A O 2 OB 2 2 Q Q 5 OC 1 Q 5 OD 2 O OU

y 3 sin 2 Drag the black dot to shift your graph in the desired direction Use the blue draggable dot to change the period Drag the orange dot to change the amplitude and or reflect with respect to the x axis The horizontal distance between the vertical dotted green lines corresponds to one period Provide your answer below 64 2TT 51 41 4 31 2 2 1 Amm 0 0 14 24 21 0 f ATT

should be injected to receive 40 units of insulin Answer 2 A formula for 1250 tablets contains 6 25 grams of diazepam How many grams of diazepam should be used in preparing 550 tablets Round to three significant figures Answer 3 If 100 capsules contain 400 mg of an active ingredient how many milligrams of the ingredient will 48 capsules contain Answer 4 A cough syrup contains 10 mg of dextromethorphan hydrobromide per 5 mL How many milligrams of the drug are contained in a 120 mL container of the syrup Answer

Compute the mean and standard deviation of the provided sample data on cost of living adjustments Then compute the z scores for the smallest and largest comparative salaries and housing adjustments and interpret your results Click the icon to view the data UCTIQUIVII Find the largest and smallest comparative salary amounts and calculate the z scores for these amounts Round to two decimal places as needed City Comparative Salary 54 578 136 024 Columbus New York Housing 9 479 Z scores Calculate the z scores for the largest and smallest housing percentages Round to two decimal places as needed City St Louis New York 0 80 2 77 is the farthest away from the Z scores 0 75 2 75 Interpret your results The z scores show that the comparative salary amount for is the farthest away from the comparative salary amount The z scores also show that the housing amount for housing amount 3 San Francisco 4 Columbus 5 Seattle 6 Louisville 7 Austin 8 Philadelphia 9 St Louis 10 Pittsburgh 11 Charleston 12 Dallas 13 Charlotte 14 DC 15 New York 16 Portland 17 18 19 20 21 22 23 24 25 26 27 28 29 105241 54578 83253 55602 57530 72048 56084 59578 60904 58072 58012 87892 136024 77349 304 1 133 2 13 72 21 22 4 5 214 479 108

hat contains 4 marbles one inge one a Determine the number of sample points in the sample space There are sample points in the sample space b Construct a tree diagram and list the sample space Construct a tree diagram Choose the correct answer below O O een one red and Click here to view diagram d Click here to view diagram b Click here to view diagram c Click here to view diagram a List the sample space Choose the correct answer below OA OG OR OY GO GR GY RO RG RY YO YG YR B O G R Y C 00 OG OR OY GO GG GR GY RO RG RR RY YO YG YR YY D OO GG RR YY c Determine the probability of selecting exactly 1 green marble The probability is Simplify your answer d Determine the probability of selecting at least 1 marble that is not green

Write sec 325 in terms of the secant of a positive acute angl Sorry that s incorrect Try again

Compute the following using a calculator cos Round your answer to two decimal places

16 Round your answer to two decimal places 9 Compute the following using a calculator cos

Given the triangle below find csc 0 Enter your answer as a ratio of sides 3 5 0

5 8 sin270 sin120 2cos360 6 30 2 2 cos 330 cos 90

The 2022 sophomore PSAT scores are normally distributed at East Richmond Hills High School with an average or mean score of 600 and a standard deviation of 25 What is the probability that a randomly chosen sophomore at East Richmond Hills earned a score less than 600 on the PSAT this year

16 Does the relation is less than for numbers have a reflex ive property consider one number A symmetric propert consider two numbers A transitive property consider three numbers

Find the amount that should be invested now to accumulate the following amount Round the answer to the nearest dollar 7400 at 6 compounded semiannually for 8 years O 4700 4611 O 4580 3840 3580

Setup an integral to find the surface area of a sphere of radius 1 using the surface areal integral formula Do not evaluate the integral Type your integral that you setup in the text box here and include your work in your File Upload

r 7 hoose the correct answer below A 14 OE 14 coordinates in three dimensional space OB 146 OC 146 0 OG OD OH

Total profit is defined as total revenue R x minus total cost C x and is given by the function P x R x C x Given R x 60x 0 3x and C x 5x 13 find each of the following a P x b R 100 C 100 and P 100 P x Type in descending powers of x R 100 C 100 S P 100

Find an approximate value of the given trigonometric function by using the figure and a calculator sin 1 2 0 5 0 0 5 a the figure Round your answer to one decimal place b a calculator Round your answer to four decimal places

2 Explain the difference if any between a CD and DC b CD and DC c CD and DC d CD and DC

23 Suppose that planes M and N intersect point A lies in both planes M and N and point B lies in both planes M and N What can you conclude regarding AB