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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

A double-pane window has a thin air space between two parallel panes of glass. If 90.0% of the incoming light is transmitted through a pane, and 10.0% is reflected, what percent of the light is
Find the indicated quantities for the appropriate arithmetic sequence.In the equation an = a1 (n − 1)d solve for n.
Write the first five terms of a geometric sequence of which the first two terms are a and b.
Find the fractions equal to the given decimals.0.0727272 . . .
Find the indicated quantities for the appropriate arithmetic sequence.Write the first five terms of the arithmetic sequence for which the second term is b and the third term is c.
Find the fractions equal to the given decimals.0.25399399399 . . .
Find the first four terms of the indicated expansions by use of the binomial series. 1 √9 - 9x
Expand and simplify the given expression.(x − 5)4
Find the indicated quantities for the appropriate arithmetic sequence.If the sum of the first two terms of an arithmetic sequence equals the sum of the first three terms, find the sum of the first
Do the term-by-term products of two geometric sequences form a geometric sequence?
Find the first four terms of the indicated expansions by use of the binomial series. √4 + x²
Find the indicated quantities for the appropriate arithmetic sequence.Find the sum of the first 100 positive integers.
Find the indicated quantities for the appropriate arithmetic sequence.Find x if 3 − x, −x, andare the first three terms of an arithmetic sequence. √9 - 2x
Find x if the sum of the terms of the infinite geometric series 1 + 2x + 4x 2 + . . . is 2/3.
Each stroke of a pump removes 8.2% of the remaining air from a container. What percent of the air remains after 50 strokes?
Expand and simplify the given expression.(3 + 0.1)4
Find the indicated quantities for the appropriate arithmetic sequence.If x, 5, and x + 8 are the first three terms of an arithmetic sequence, find the sum of the first 20 terms.
Find the sum of the terms of the infinite series 1 + 2x + 3x2 + 4x3 + . . . for |x|< 1.
From 2015 to 2020, Canada had a 0.950% average growth rate of population, compounded annually. If the population in 2015 was 36.0 million, what was the 2020 population?
Expand and simplify the given expression.(x2 + 4)5
Solve the given problems involving factorials.Using a calculator, evaluate(a) 17! + 4!, (b) 21!, (c) 17! × 4!,(d) 68!.
An electric current decreases by 12.5% each 1.00 μs. If the initial current is 3.27 mA, what is the current after 8.20 μs?
Expand and simplify the given expression.(3n1/2 − 2a)6
Find the first four terms of the appropriate expansion. 12 ( 4 − y )"
Solve the given problems involving factorials.Using a calculator, evaluate (a) 8! − 7!, (b) 8!/7!, (c) 8! × 7!,(d) 56!.
Find the indicated quantities for the appropriate arithmetic sequence.If x, x + 2y, 2x + 3y, . . . is an arithmetic sequence, express S100 in terms of x.
Find the first four terms of the appropriate expansion. (p². (9
A copying machine is set to reduce the dimensions of material copied by 10%. A drawing 12.0 cm wide is reduced, and then the copies are in turn reduced. What is the width of the drawing on the sixth
Find the first four terms of the appropriate expansion.(a + 3e)10
Find the indicated quantities for the appropriate arithmetic sequence.The sum of the angles inside a triangle, quadrilateral, and pentagon are 180°, 360°, and 540°, respectively. Assuming this
Solve the given problems involving factorials.Show that (n + 1)!/(n − 2)! = n³ n for n ≥ 2.
How much is an investment of $250 worth after 8 years if it earns annual interest of 4.8% compounded monthly? [4.8% annual interest compounded monthly means that 0.4% (4.8% 12) interest is added each
Solve the given problems involving factorials.Show that n! = n × (n − 1)! for n ≥ 2. To use this equation for n = 1, explain why it is necessary to define 0! = 1 (this is a standard definition
Find the indicated terms by use of the following information. The r + 1 term of the expansion of (a + b)n is given byThe term involving y6 in (x + y)10 r factors n(n-1)(n-2)(n-r+1) r! an-rbr
Find the indicated terms by use of the following information. The r + 1 term of the expansion of (a + b)n is given byThe term involving b5 in (a + b)8 r factors n(n-1)(n-2)(n-r+1) r! an-rbr
Find the indicated terms by use of the following information. The r + 1 term of the expansion of (a + b)n is given byThe fifth term of (2x − 3b)12 r factors n(n-1)(n-2)(n-r+1) r! an-rbr
Find the indicated quantities for the appropriate arithmetic sequence.A person begins an exercise program of jogging 10 min each day for the first week. Each week thereafter, the person must increase
A chemical spill pollutes a stream. A monitoring device finds 620 ppm (parts per million) of the chemical 1.0 mi below the spill, and the readings decrease by 12.5% for each mile farther downstream.
Find the indicated quantities for the appropriate arithmetic sequence.A beach now has an area of 9500 m2 but is eroding such that it loses 100 m2 more of its area each year than during the previous
Measurements show that the temperature of a distant star is presently 9800°C and is decreasing by 10% every 800 years. What will its temperature be in 4000 years?
Find the indicated quantities for the appropriate arithmetic sequence.During a period of heavy rains, on a given day 110, 000 ft3/s of water was being released from a dam. In order to minimize
Find the first four terms of the appropriate expansion.(2s)2 −3/2t−1)14
A swimming pool has an automatic cleaning system. When debris has entered the pool, the system can remove 15% of it in an hour. How long does it take for this system to remove 80% of the debris from
Find the first four terms of the indicated expansions by use of the binomial series.(1 + x)12
Find the indicated terms by use of the following information. The r + 1 term of the expansion of (a + b)n is given byThe sixth term of (√a − √b)14 r factors n(n-1)(n-2)(n-r+1) r! an-rbr
Find the first four terms of the indicated expansions by use of the binomial series. √1 + x² 2
Find the indicated quantities for the appropriate arithmetic sequence.At a logging camp, 15 layers of logs are so piled that there are 20 logs in the bottom layer, and each layer has 1 less log than
The strength of a signal in a fiber-optic cable decreases 12% for every 15 km along the cable. What percent of the signal remains after 100 km?
Find the first four terms of the indicated expansions by use of the binomial series. √1-a² १२
Find the first four terms of the indicated expansions by use of the binomial series.(1 − 2x)10
A series of deposits, each of value A and made at equal time intervals, earns an interest rate of i for the time interval. The deposits have a total value ofA(1 + i) + A(1 + i)2 + A(1 + i)3 + . . . +
The power on a space satellite is supplied by a radioactive isotope. On a given satellite, the power decreases by 0.2% each day. What percent of the initial power remains after 1 year?
Find the indicated quantities for the appropriate arithmetic sequence.In order to prevent an electric current surge in a circuit, the resistance R in the circuit is stepped down by 4.0 Ω after each
Without expanding, evaluate(2x −1)3 + 3(2x −1)2 (3 − 2x) + 3(2x −1)(3 − 2x)2 + (3 − 2x)3
A new car that was purchased for $40,000 depreciates by 10% each year. Find the value of the car (to the nearest dollar) when it is 10 years old.
Find the first four terms of the indicated expansions by use of the binomial series.(4 − 4√x)−1
Find the first four terms of the indicated expansions by use of the binomial series. √1 +264
Add up the coefficients of the first few powers of (a + b)n by using Pascal’s triangle. Then by noting the pattern, determine the sum of the coefficients of (a + b)79 .
Find the indicated quantities for the appropriate arithmetic sequence. A car depreciates $1800 during the first year after it is bought. Each year thereafter it depreciates $150 less than the year
Find the indicated quantities for the appropriate arithmetic sequence.A bank loan of $8000 is repaid in annual payments of $1000 plus 10% interest on the unpaid balance. What is the total amount of
Explain why n! ends in a zero if n > 4.
If you decided to save money by putting away 1¢ on a given day, 2¢ one week later, 4¢ a week later, and so on, how much would you have to put away 6 months (26 weeks) after putting away the 1¢?
How many direct ancestors (parents, grandparents, and so on) does a person have in the ten generations that preceded him or her (assuming that no ancestor appears in more than one line of descent)?
Find the indicated quantities for the appropriate arithmetic sequence. The sequence of ships’ bells is as follows: 12:30 a.m. one bell is rung, and each half hour later one more bell is rung than
A professional baseball player is offered a contract for an annual salary of $5,000,000 for 6 years. Also offered is a bonus (based on performance) of either $400,000 each year, or a 5.00% increase
Expand [(a + b) + c]3 using the binomial theorem. Group terms as indicated.
Find the first four terms of the indicated expansions by use of the binomial series.(2 − 4x)−3
A mechanical system has 4 redundant engines, each with a 95% probability of functioning properly. When the binomial theorem is used to expand (0.95 + 0.05)4 , the resulting terms give the
Approximate √6 to hundredths by noting thatand using four terms of the appropriate binomial series. √6 √4(1.5) = 2√1 + 0.5 =
Approximate 3√10 by using the method of Exercise 51.Data from Exercises 51Approximate √6 to hundredths by noting thatand using four terms of the appropriate binomial series. √6 √4(1.5) =
Find the indicated quantities for the appropriate arithmetic sequence.When a stone is dropped from the edge of a cliff at the Grand Canyon, it falls 16 ft during the first second, 48 ft during the
If 85% of an aspirin remains in the bloodstream after 3.0 h, how long after an 80mg aspirin is taken does it take for there to be 35 mg in the bloodstream?
Find the first four terms of the indicated expansions by use of the binomial series.(1 + 4x)−1/4
In Exercise 49, find the probability that at least one engine functions properly (the sum of the terms containing 0.954 , 0.953, 0.952, and 0.951). Round to the third decimal place.Data from
Find the indicated quantities for the appropriate arithmetic sequence. A college graduate is offered two positions. A computer company offers an annual salary of $42,000 with a guaranteed annual
Find the indicated quantities for the appropriate arithmetic sequence.In preparing a bid for constructing a new building, a contractor determines that the foundation and basement will cost $605,000
Find the sum of the first 1000 positive even integers.
A certain tennis ball was dropped from a height of 100 ft. On each bounce, the ball reached a height equal to 55% of the previous height. Find the height of the ball (to the nearest hundredth) after
Write down several terms of a general geometric sequence. Then take the logarithm of each term. Explain why the resulting sequence is an arithmetic sequence.
Solve the given problems by use of an appropriate sequence or series. All numbers are accurate to at least two significant digits.Find the sum of the first 300 positive odd integers.
Find the indicated quantities for the appropriate arithmetic sequence.A person has a $5000 balance due on a credit card account that charges 1% interest per month on the unpaid balance Bn. Assuming
Do the squares of the terms of a geometric sequence also form a geometric sequence? Explain.
Solve the given problems by use of an appropriate sequence or series. All numbers are accurate to at least two significant digits.What is the fifth term of the arithmetic sequence in which the first
In the theory associated with the magnetic field due to an electric current, the expressionis found. By expanding (a2 + x 2)−1/2, find the first three nonzero terms that could be used to
A company purchases a piece of equipment for A dollars, and the equipment depreciates at a rate of r each year. Its value V after n years is V = A(1 − r)n . Expand this expression for n = 5.
Find the indicated quantities for the appropriate arithmetic sequence.Derive a formula for Sn in terms of n, a1, and d.
If a1, a2, a3, . . . is an arithmetic sequence, explain why 2a1 , 2a2 , 2a3 , . . . is a geometric sequence.
Find three consecutive numbers in an arithmetic sequence such that their sum is 15 and the sum of their squares is 77.
In finding the rate of change of emission of energy from the surface of a body at temperature T, the expression (T + h)4 is used. Expand this expression.
In the theory related to the dispersion of light, the expressionarises. (a) Let x = λ02/2 and find the first four terms of the expansion of (1 − x)−1.(b) Find the same expansion by using
Find the indicated quantities for the appropriate arithmetic sequence.A harmonic sequence is a sequence of numbers whose reciprocals form an arithmetic sequence. Is a harmonic sequence also an
The numbers 8, x, y form an arithmetic sequence, and the numbers x, y, 36 form a geometric sequence. Find all of the possible sequences.
For a geometric sequence, is it possible that a3 = 6, a5 = 9, and a7 = 12?
Find the indicated quantities for the appropriate arithmetic sequence.Show that the sum of the first n positive integers is 1/2n(n + 1)
For f(x) = ab x , is f(1), f(2), f(3) an arithmetic sequence or a geometric sequence? What is the common difference (or common ratio)?
A cubic Bezier curve is defined by four points (P0 through P3) from the parametric equations given by (1 − t)3P0 + 3(1 − t)2 tP1 + 3(1 − t)t2P2 + t3P3, 3 where 0 ≤ t ≤ 1. Note that the
Find three consecutive numbers in a geometric sequence such that their product is 64 and the sum of their squares is 84.
Find the indicated quantities for the appropriate arithmetic sequence.Show that the sum of the first n positive odd integers is n2.
Find a formula with variable n of the arithmetic sequence with a1 = −5, an + 1 = an −3, for n = 1, 2, 3, . . . .
Solve the given problems by use of an appropriate sequence or series. All numbers are accurate to at least two significant digits.Approximate the value of √30 by noting thatand using three terms of

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