MATH

(Nos 4-6) Find the centroid of each of the areas bounded by the following curves. 4. 2x + y = 6, x=0, y=0

a. (1,2)

(Nos 4-6) Find the centroid of each of the areas bounded by the following curves. 6. 𝑦 = 𝑥 2 − 4, 𝑦 = 2𝑥 − 𝑥 2

a. (1/2 , -3/2)

Find the differential equation whose general solution is y = 𝑐1𝑥 + 𝑐2𝑒 𝑥 .

a. (x - 1)y" - xy' + y = 0

(Nos. 36-40) Obtain the 3rd term of the Maclaurin's series of the following functions. 39. ln cos x

a. - (1/45) x6

(1⁄𝑖)^2509

a. - i

Find the vertical shift of the graph of y = 5cos2x.

a. 0

Find the Laplace transform of f(t) = cos2 (3t)

a. 1/2 ( 1/𝑠 + 𝑠/𝑠^2+36 )

From point O on the ground of a square courtyard of area 160,000 ft2 , the angles of elevation of three flagstaffs of equal heights at three consecutive corners of the yard at A, B, and C are 45°, 60° and 60°, respectively. Find the height of each flagstaff.

a. 114.31 m

How many 3 digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9 which are divisible by 5 and none of the digits is repeated?

a. 20

What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the yaxis?

a. 2ydx - xdy = 0

Find the period of the graph of y = sin x

a. 2π

Determine the area below f(x) = 3 + 2x - x 2

a. 32/3

Jacob wants to go to a job interview in another city. He can choose from 4 local bus services or 3 taxi services to reach the other city. From there, he can choose from 3 local bus services or 2 taxi services to reach the place of his interview. How many possible routes/ways Jacob can travel to his destination?

a. 35

The area enclosed by the curve x2 + y2 = 25 is revolved about the line x - 10 = 0. Find the volume generated.

a. 4,934.80

Determine the mean

a. 43.84

Determine the mean.

a. 43.84

A rectangle having a width of 2 units and height of 4 units has its centroid at (10, 12). Find the volume generated when the rectangle is revolved about the y-axis.

a. 502.65

Solve for the inverse laplace transform of 7s+15/(𝑠^2+2)

a. 7 cos(√2𝑡) + 15/√2 sin (√2𝑡)

Determine the distance of the point from the center of the bigger pulley where the belt will cross when crossconnected.

a. 91.3 cm

Solve the equation (cos 2y - 3x2 y 2 ) dx + (cos 2y - 2xsin 2y - 2x3 y) dy = 0.

a. sin 2y + 2x cos 2y - 2x3 y 2 = c

Find the horizontal shift of the graph of y = 3 sin(6x-π ) + 10

a. π/6

Solve for the particular solution of dy/dx=6y2 x having an initial condition y(1) = 1/25

a. 𝑦 = 1 /(28−3𝑥^2)

(Nos. 36-40) Obtain the 3rd term of the Maclaurin's series of the following functions. 40. 1/√(4−𝑥)

b. (3/256) x2

Find the amplitude of the graph of y=sin x

b. 1

There are 8 men and 10 women and you need to form a committee of 5 men and 6 women. In how many ways can the committee be formed?

b. 11760

Situation 3: (Nos. 20-22) A certain radioactive element follows the "law of exponential change" and has a "half-life" of 38 hours. 20. How long it takes for 90% of the radioactivity of the element to be dissipated?

b. 126 hrs

If the area enclosed by the ellipse on the first and second quadrants is revolved about the x-axis, what is the volume generated?

b. 150.80

An airplane travels 360 miles in two hours with the wind and flying back the same route, it took hours against the wind. Find the velocity of the wind.

b. 40 mph

Compute the length of belt if both pulleys will rotate in the same direction.

b. 446.0 cm

Find the amplitude of the graph of y=5cos3x

b. 5

How many possible combo deals does Fluffy have?

b. 8

Find the area of the region bounded by y = x2 + 2; y = sin x; x = -1; x = 2.

b. 8.04355

. Find the moment of inertia with respect to the x-axis of the region bounded by the parabola x2 = 8y, the line y = 2, and the y-axis a. 8.4

b. 9.14

What is the equation of the diameter of the ellipse which bisects all chords having a slope of 2?

b. 9x + 32y = 0

If a = b, then b = a. This property of real numbers is known as

b. symmetric property

What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the xasis?

b. y dx - 2x dy = 0

Solve for the inverse Laplace transform of F(s) = 2𝑠+3/𝑠^2+4𝑠+13

b. 𝑒 ^2𝑡 (2𝑐𝑜𝑠3𝑡 − 1/3 𝑠𝑖𝑛3𝑡)

(Nos. 36-40) Obtain the 3rd term of the Maclaurin's series of the following functions. 38. sec x

c. (5/24) x4

(Nos 4-6) Find the centroid of each of the areas bounded by the following curves. 5. y = 2x + 1, x + y = 7, x=8

c. (6,7)

Find the differential equation of the family of circles with center at (h, k)

c. (x - h) dx + (y - k) dy = 0

A motorboat goes 3 miles upstream in the same time required to go 5 miles downstream. If the rate of flow of the river is 3 miles per hour, find the speed of the motorboat in still water.

c. 12 mph

A perfumer wishes to blend perfume valued at $4.10 an ounce with perfume worth $2.50 an ounce to obtain a mixture of 40 ounces worth $3.00 an ounce. How much of the $4.10 perfume should he use?

c. 12.5 oz.

In how many different ways can the letters of the word 'RUMOUR' be arranged?

c. 180

Find the Laplace transform of f(t) = sin (2t) cos (2t)

c. 2 /(𝑠^2 + 16)

Find the amplitude of the graph of y = 3 sin (6x-π)

c. 3

The area bounded by the curve y = sin x from x = 0 to x = is revolved about the x-axis. What is the volume generated?

c. 4.93

What is the length of arc in the first quadrant of the ellipse?

c. 5.55

Find the period of the graph of y = 3 sin (6x-π)

c. 60°

How many 3-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?

c. 720

A bacterial population is known to have a rate of growth proportional to the population itself. If between noon and 2 P.M. the population triples, at what time, no controls being exerted, should the population become 100 times what it was at noon.

c. 8:23 PM

The angle of elevation of the top point D of a tower from A is 23°30'. From another point B the angle of elevation of the top of the tower is 55°30'. The points A and B are 217.45 m apart and on the same horizontal plane as the foot (point C) of the tower. The horizontal angle subtended by A and B at the foot of the tower is 90°. Find the height of the tower CD.

c. 90.59 m

Two sides of a triangle are 50 m and 60 m long. The angle included between these sides is 30°. What is the exterior angle opposite the longest side?

c. 93.74°

If a = b and b = c, then a = c. This is the _______property of real numbers

c. transitive

Solve the equation y dx + (3x - xy + 2) dy = 0

c. xy3 = 2y2 + 4y + 4 + cey

The equation y2 = cx is the general solution of :

c. y' = y/2x

(Nos. 36-40) Obtain the 3rd term of the Maclaurin's series of the following functions. 36. 1/2 (𝑒 𝑥 + 𝑒 −𝑥 )

c. 𝑥^4/4!

Find the differential equation of the family of circles with radius R

c. 𝑹 = (𝟏+𝒚^𝟐 )^𝟑⁄𝟐 𝒚′′

(Nos. 36-40) Obtain the 3rd term of the Maclaurin's series of the following functions. 37. tan x

d. (2/15) x5

What is the differential equation of the family of lines passing through the fixed point (h, k)?

d. (y - k) dx - (x - h) dy = 0

. Find the horizontal shift of the graph of y = 5cos2x

d. 0

Find the moment of inertia of the region bounded by the curves x2 = 8y, x - 4 = 0, and the x-axis about the x-axis.

d. 1.52

Find the vertical shift of the graph of y = 3 sin(6x-π ) + 10

d. 10

Determine the area of the region bounded by = 𝑥√𝑥 2 + 1 ; 𝑦 = 𝑒 − 1 2 𝑥 ; 𝑥 = −3 ; and the y - axis

d. 17.171

Determine the standard deviation.

d. 22.81

Cheri went to a dress shop to purchase a single blue dress. The shop has 14 plain blue dresses, 13 plain red dresses, 10 blue embroidered dresses, and 5 red embroidered dresses. How many options does Cheri have?

d. 24

Determine the percentage of radioactivity that remains after 76 hours

d. 25 %

Find the moment of inertia of an area bounded by the curve x2 = 8y, the line x - 4 = 0, and the x-axis about the y-axis.

d. 25.60

Compute the length of belt if the belt will be cross-connected to make the pulleys rotate in opposite directions.

d. 465.2 cm

A thermometer reading 18°C is brought into a room where the temperature is 70°C. One minute later, the thermometer reading is 31°C. Determine the thermometer reading 5 minutes after it is brought into the room.

d. 58 °C

. In a murder investigation, a corpse was found by a detective at exactly 8 P.M. Being alert, the detective also measured the body temperature and found it to be 70 °F. Two hours later, the detective measured the body temperature again and found it to be 60 °F. If the room temperature is 50 °F, and assuming that the body temperature of the person before death was 98.6° F, at what time did the murder occur?

d. 5:30 PM

A Toyota Land Cruiser drives east from point A at 30 kph. Another car, Ford Expedition, starting from B at the same time, drives S30°W toward A at 60 kph. B is 30 km from A. How fast in kph is the distance between two cars changing after 30 minutes?

d. 60 kph

How many ounces of pure silver must be added to 100 ounces, 40% pure, to make an alloy which is 65% pure silver?

d. 71 oz.

A statement of equality between two ratios.

d. proportion