In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?

In two regular polygons with the same number of sides, find the length of a side of the smaller if the lengths of apothems are 20 and 50 and a side of the larger has length 32.5.

Mines in a harbor during wartime are so placed that ship going to a certain port within the harbor, to avoid hitting them must follow a path which is at all times equidistant from a point 9 km north of another port and a line whose bearing is due east. One point of this line is 3 km south of the port. Considering the port as the origin, find the equation of the path of the ship.

If the sum of the angles A + B + C = 180° and tan A + tan B + tan C = x, find the equivalent value of (tan A) (tan B) (tan C).

A rectangular prism of metal having dimensions 4.3 cm by 7.2 cm by 12.4 cm is melted down and recast into a frustum of a square pyramid, 10% of the metal being lost in the process. If the ends of the frustum are squares of side 3 cm and 8 cm respectively, find the thickness of the frustum.

A quadrilateral ABCD is inscribed in a circle. The angle ADC = 75° and angle ACD = 45°. If the arc BC is only one-half of arc CD, find the value of angle BCA.

Given a circle of radius 4 and center at (6,8). Find the locus of the center of all circles that is tangent to the x-axis and the given circle.

In hilly or mountainous terrain, it is often difficult to determine heights. It may be impossible to find an expanse of level ground from which measurements can be taken, and sighting may be limited. Mountain peaks may be visible only from certain places. A geologist took at two points, B and C, 928 m apart. The angle of elevation from B to the top of the mountain at A is 47°10'. The angle of elevation from C to B is 8°40', and the angle of elevation from C to A is 32°30'. It is known that the elevation at C is 1621 m to the nearest meter, what is the elevation at A?

The probabilities that three men hit a target are 1/6, ¼, and 1/3, respectively. Each shoot once at a target. If only one of them hits the target, find the probability that it was the first man?

A total of 300 people composed of students and teachers in a certain school are asked in a poll. The answers of the poll are either they are to agree, disagree or just be neutral to an issue being asked. From the students, there are 132 who agreed, 78 who disagreed and 54 are neutral. There are 5 who agree, 14 who disagreed and 17 are neutrals from the teachers. If a teacher and a student are randomly selected, what is the probability that they either agree or neutral?

A steel storage tank for methane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with hemisphere attached to its ends. Express the volume (in cubic feet) of the tank as function of the tank radius.

An elliptical lot has a major axis of 300 m and a minor axis of 250 m. If the cost of fencing is Php 1,500 per meter, what is the total cost of fencing the lot?

The scale of an Ordinance Survey Map is 1: 2500. A circular sports field has a diameter of 8 cm on the map. Calculate its area in hectares.

Find the value of θ if versed sin of θ = 0.148.

Twenty-eight persons can do a job in 60 days. They all start complete. Five persons quitted the job at the beginning of the 16th day. They were reinforced with 10 persons at the beginning of the 45th day. How many days was the job delayed?

A right regular hexagonal prism is inscribed in a right circular cylinder whose height is 20 cm. The difference between the circumference of the circle and the perimeter of the hexagon is 4 cm. Determine the volume of the prism?

Find the distance between the foci of the curve 9x2+25y2−18x+100y−116=0.

A flagpole 3 m high stands at the top of a pedestal 2 m high located at one side of the pathway. At the opposite side of the pathway directly facing the flagpole, the flagpole subtends the same angle as the pedestal. What is the width of the path way?

Given a regular hexagon with consecutive corners ABCDEF. If the bearing of side AB is N 25° E, what is the bearing of side FA?

In triangle ABC, AB = 30 cm, BC = 36 cm, and AC = 48 cm. The perpendicular bisectors of the sides intersect at point P. How far is P from side BC?

A rectangular swimming pool is 12 meters long and 8 meters wide. A tile border of uniform width is to be built around the pool, using 102 square meters of tile. The tile is from a discounted stock (so no additional materials are available), and all 120 square meters are to be used. How wide should the border be?

There are 6 people lining up to pay telephone bills. If two persons don't want to follow each other, how many different lineups is possible?

A line has an equation of x + 5y + 5 = 0. Find the equation of the line through (3,1) that makes an angle of 45° clockwise from the line that is perpendicular to the line x + 5y + 5 = 0 at that point.

A spherical ball was completely immersed into an inverted right circular cone full of water. After the ball was removed, it was found out that the water surface had dropped 5.7258 cm below the top of the cone. The diameter of the cone is 12 cm and its altitude is 36 cm. Which of the following gives the diameter of the spherical ball?

A conical tank 1 m diameter on top and 2 m deep is filled with water. Find the weight of water in the tank if the unit weight of water is 9.81 kN/m3.

Given a triangle whose sides are 10 cm, 24 cm, and 26 cm. Find the perimeter of a smaller triangle that is similar to the given triangle whose shortest side is 6 cm.

What is the slope-intercept for the line 3x - 2y + 8 = 0?

A card is drawn out from a deck of 52 playing cards. Determine the probability that a red or a face card is drawn.

The sum of two angles is 1600 mils and their difference is 40 grads. Find the value of the bigger angle in degrees.

A 20-inch belt connects a pulley with a very small shaft on a motor. The distance between the shaft and the large pulley is 4 inches. Assuming the shaft as a single point, find the radius of the large pulley to the nearest tenth of an inch. The length of belt in contact with the pulley is 8.5 inches.

Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the man who shoots first?

A container whose ends are isosceles trapezoid is 10 m long. The lower base is 2 m the upper base is 6 m and the depth is 4 m. It contains 100,000 liters of liquid. How many spherical balls of 0.48 m diameter should be placed in the container so that the liquid surface is at the top edge of the latter?

Find the area enclosed by the curve whose equation in polar form is given by r = 2 sin θ + 2 cos θ.

A triangle is inscribed in a circle of radius 10 cm. One side of the triangle is 20 cm long and its area is 96 cm2. What is the perimeter of the triangle?

The wagon wheel has a diameter of 28 inches and an angle of 30° between the stokes. What is the length of the arc s between the adjacent stokes?

The boat travels downstream in two-thirds the time as it does upstream. If the speed of the river current is 8 kph, determine the velocity of the boat in still water?

A triangular plot of land has sides, a = 22.4 m, b = 25.2 m, and c = 11 m. Find the measures of the largest angle?

The equation of a circle is x2+y2+2ky=0. What is the value of k when the length of the tangent from (5,4) to the circle is 1?

A horizontal cylindrical tank has a radius of 600 mm and a length of 5 m. The tank is 7/8 full. Determine the depth of water in the tank when it is in horizontal position?

In a large city, streets are run north and south, and avenues run east and west. Streets and avenues are 850 feet apart. The city plans to construct a straight freeway from the intersection of 25th street and 8th avenue to the intersection of 115th street and 64th avenue. How long will the freeway be?

Given triangle ABC, point E is along side AC, and point D is along the side BC. Line BE and AD are the angle bisectors of angle B and A respectively. If the angle BEC = 75°and angle ADC = 84°, find angle ACB.

10 business executives and 7 chairmen meet at a conference. If each business executive shakes the hand of every other business executive and every chairman once, and each chairman shakes the hand of each of the business executives but not the other chairmen, how many handshakes would take place?

Given a cube of side 1 m. Find the volume of the cone inscribed in this cube, with base of cone on one side of the cube.

The scale of a certain map is 1: x. A lot having an area of 640 square meters is represented by an area of 25.6 cm2 on the map. What is the value of x?

A sphere having a volume of 2000 cm3 is cut by a plane 4 cm from its center. What is the curved surface area of the smaller segment cut by the plane?

Points A, B, and C are on a circular track with AB as a diameter of the track. The angle of elevation of the top of a vertical pole standing at A as observed from B is 25°. If the horizontal angle subtended by BC at A is 30°, what is the angle of elevation of the top of the pole as observed from C?

Side AB and AD of a corner lot ABCD are on two streets intersecting at an angle of 75°. AB = 12 m while AD =16 m. Side BC is perpendicular to AB and CD is perpendicular to AD. Find the area of the lot.

If M is the midpoint of PQ, find the coordinates of M if the coordinates of P and Q are P (3,4) and Q (5,8).

A poultry farm has only chickens and pigs. When the manager of the poultry counted the heads of the stock in the farm, the number totaled up to 200. However, when the number of legs was counted, the number totaled up to 540. How many chickens were there in the farm?