Sections 1.1-1.3 and 10.1-10.4

During vector addition, how is the angle of the resultant calculated?

The value of a trigonometric function calculated from the dimensions of the resultant vector is used in the inverse tangent function to calculate the resultant vector angle. Inverse trigonometric ratios, in general, can be used o calculate angles.

How do you generate a third vector or a scalar by multiplication?

To generate a scalar like work, we multiply the magnitudes of the two vectors of interest (force and displacement) and the cosine of the angle between them

True of False: Total distance traveled can never be less than the total displacement

True. Displacement considers the most direct route between two points. Distance will always be equal or larger larger in magnitude than the displacement.

Determine the volume of a cylinder with a radius that is measured as 7.45m and a height of 8.323m. (Note: Use 3.14159 as pi), and round the answer to the correct number of significant digits).

V = A base x h = (pi r^2) x h = (3.14159)( 7.45 ^2) x (8.323) 1451.249 Because we only need the lowest amount of significant figures, which is this case is the radius (two significant figures) last number can be ignored, so the final answer will be 1500 or 1.5 x 10^3

When calculating the sum of vectors A and B (A + B) we put the tail of B at the tip of A. What would be the effect of reversing this order (B + A)?

Vector addition, is a commutative function. The resultant of A + B is the same and B + A, so there will be no difference between the resultants.

When calculating the difference of vectors A and B (A - B) we invert and put the tail of this new vector at the tip of A. What would be the effect of reversing the order (B - A)?

Vector subtraction, like vector multiplication, is not a commutative function. The resultant of A - B has the same magnitude as B - A, but is oriented in the opposite direction

What are the units of Power?

Watt (W) (kg x m^2 / s^3)

Which of the following quantities is NOT a vector? A. Velocity B. Force C. Displacement D. Distance

D. A vector is characterized by both magnitude and direction. From the given answer choices, all are vectors except for distance. Distance is a scalar because it has only a numerical values and lacks direction.

Ina certain rigid container, pressure and temperature are directly proportional. If the pressure is changed from 540 torr to 180 torr via a temperature change, by what factor had the temperature changed? A. 360 B. 3 C. 1 D. 1/3

D. Ina direct relationship, a change in one of the variables will be associated with a proportional change in the other. Because pressure was multiplied by 1/3, the temperature must also be multiplied by 1/3. Note that the fractional relationships can only be used with temperatures in Kelvin.

Which of the following equations is incorrect? A. A^3 x B^3 = (AB)^3 B. A^5 / A^7 = A^-2 C. (A^0.5)^4 + A^2 = 2A^2 D. (A^3)^2 = A^9

D. Raising an exponent to another exponent requires multiplying the exponents. Thus (A^3)^2 = A^6

Which of the following relationships is INCORRECT? A. Isin() x cos()I < Isin()I + Icos()I B. Sin () / cos() = tan () C. tan 90 is undefined D. sin () = sin (90 - ())

D. sin () does not equal sin (90- ()), although sin() = cos(90 -()). The other statements must all be true. Because sine and cosine values are always between -1 and 1, the product of sine and cosine will always have the a magnitude less than 1. The sum of the absolute value of sine and the absolute value of cosine. on the other hand will always be greater than 1. Therefore, Choice (A) can be eliminated. Because sine is the ratio of opposite to hypotenuse and cosine is the ratio of adjacent to hypotenuse, the quotient between the two ratios is the ratio of opposite to adjacent, or the tangent of the angle. Therefore, Choice (B) can be eliminated. By the same logic, because sin90 = 1 and cos90=0, tan90 is undefined, eliminating choice C.

Estimate the value of 15.4 / 3.80

Estimations in divisions should be made by shifting both numbers in the same direction. It is often easier to adjust the divisor first to simplify calculations, If we round the divisor up to 4, we should round the dividend up accordingly. In this case, it makes sense to round the dividend up to 16- which is not only a whole number, but also a multiple of 4. Our estimate is 16 / 4 = 4. Despite being a rough adjustment, we are still very close to the true value of 4.05

True or False: Only angles in right triangles have characteristic values of the trigonometric functions?

False. While calculating the values of Sine, Cosine, and Tangent is more complicated in a triangle that does not contain a right angle, all possible angles do still have characteristic trigonometric values.

True or False: If C = A x B, where A is directed toward the right side of the page and B is directed to the top of the page, then C is directed mid-way between A and B at the angle 45 angle.

False: this would be true of an addition problem in which both vectors have equal magnitude, but it is never true for vector multiplication. To find the direction of C, we must use the right-hand rule. If the thumb points in the direction of A, and the fingers point in the direction of B, then our palm, C point into the page.

Describe the process for converting a number into scientific notation. What values are possible for the significand?

First determine which digits are significant, as these will be preserved in scientific notation. Then, move the decimal point until the significand is greater than or equal to 1 and less than 10. Finally, determine what power of 10 is necessary for multiplication to restore the original number.

How do you generate a third vector like torque?

First we determine the magnitude and direction and we multiply the two vectors (force and lever arm) and the sine between the two vectors. This is known as the cross product

What does it mean for two variables to have a direct relationship? Direct: Inverse:

In direct relationships, as one quantity increases, the other also increases in proportion. In inverse relationships, as one quantity increases, the other decreases in proportion.

When rounding two numbers containing decimals, in which direction(s) should each number go for multiplication? For division?

In multiplication, adjust the two decimal in opposite direction (One rounded up and one rounded down). In division, adjust the two decimals in the same direction (both either rounded up or both rounded down).

Each of the three methods for solving systems of equations discussed in this chapter solve for one variable, and then use this value to solve for the other. How does each method solve for the first variable? Substitution: Setting equations equal: Elimination:

In substitution, solve one equation for one variable in therms of the other; then, substitute this expression into the other equation. In setting equations equal to (a modified version of substitution), solve both equations for the same variable and set them equal to each other. In elimination, multiply or divide one (or both) equations; then add or subtract the equations to eliminate one of the variables.

How are conversions between metric prefixes accomplished?

conversions between metric prefixes are accomplished by multiplication or division by the relevant power of ten.

How many significant figures are int he following numbers? 34,600. 0.0003201 1.10 525,600

34,600. ------- 5 sig figs (34,600) .0003201------ 4 sig figs (3201) 1.10------------ 3 sig figs (1.10) 525,600------- 4 sig figs (525,6)

Estimate the square root of 392

392 is between Sqr 361 and Sqr 400, so the value is between 19 and 20. we can also identify this radical: Sqr 392 = Sqr4 x sqr49 x sqr2 about 14 x1.4 = 19.6 actual = 19.8

what is the relationship between instantaneous velocity and instantaneous speed?

Instantaneous speed is the magnitude of the instantaneous velocity vector.

A cars speedometer registers a speed of 35 miles per hour. What was its speed in meters per second?

first convert distance measurements, being careful to cancel them out by arranging numerators to denominators. (35miles/hr) x (5280feet/ 1 mile) x (12in / 1 foot) x (2.54cm / 1 in) x (1m / 100 cm) = 56,327 m/hour Convert hours to seconds: 1 hour has 3600 seconds 56,327 Meters / hour (1 hour/ 3600s) = 15.6 Meters per second

The ejection fraction is the portion of the left ventricular volume expelled with each contraction of the heart. A patient is known to have an ejection fraction of 0.7, a cardiac output of 5 L/min, and a heart rate of 80 beats/ minute. What is the volume of the left ventricle in this person?

A formula was not provided in this question, but we can recognize that the desired answer is a volume. We can start with the cardiac output and heart rate to determine the volume ejected per beat: (5 L/min) / (80 beats/ min) = 0.625 L/beat This question also explains that only 70 percent of the volume of the left ventricle is expelled per heartbeat. from this, we can determine the volume of the ventricle. 0.0625L/beat = 0.7 x ventricular volume/ beat 0.0625/0.7 L = ventricular volume ~~ 0.09 L

How is a scalar calculated from the product of two vectors?

A scalar is calculated from two vectors by using the dot product.

How is a vector calculated?

A vector is calculated by using the cross product

The value of 200^0.25 is closes to? A. 4 B. 14 C. 50 D. 800

A. The fourth root of a number, or a number raised to the one quarter power, is the root of the square root of that number. 200^0.25 = 200 ^(1/2 x 1/2) = Sqr(sqr200) = 4sqr200 The square root of 200 should be a bit larger than 14 (14^2 = 196); therefore, the fourth of 200 should be a bit less than 4.

what is the minimum value of 2 cos (angle) -1? A. -3 B. -2 C. -1 D. 0

A. The minimum value of the cosine function is -1 (cos 180 = -1). Therefore, the minimum value of 2 cos () - 1 is 2 x (-1)(-1) =-3

Which of the following would be the most appropriate set up for estimating the value of 3.6 x 4.85 for questions in which answer choices differ by a small margin? A. 3.5 X 5 B. 3.5 X 4.5 C. 4X 4 D. 4 X 5

A. When estimating the product of two numbers, it is best to round up while rounding the other down, as in Choice (A). Choices (B) and (D) each round both numbers in the same direction, which would increase the amount of error in the answer. Choice (C) round the numbers in opposite direction, but the degree of rounding is significantly larger than in Choice (A) and too extreme for answer choices that differ by small amounts.

The rate of a reaction is calculated as a change in concentration per time. What are the units of the rate constant, k, in a reaction that is second order overall with respect to one species? (Note: A second-order reaction of this type has a rate law with the form rate=k(A)^2, where A is the concentration of the species.) A. 1/s B. M/s C. L/mol x s D. L^2 /mol^2 x s

A. According to the question stem, the rate of a reaction is measured as a change in concentration over time, and thus has the units M/s, where M (molarity) is measured in moles per liter. However, the rate of the reaction is equal to a rate constant times the concentrations of certain reactants squared. In this case, we known the units of everything except the rate constant and must solve for its units. rate= k(A)^2 M/s= k(M)^2 -> k=1/M x s = L/mol x s

How can the value of a natural logarithm be converted to the value of a common logarithm? A. The natural logarithm is divided by a constant. B. A constant is added to or subtracted from the natural logarithm C. The natural logarithm is raised to an exponent D. The inverse of the natural logarithm is taken

A. The relationship between the natural logarithm of a number and the common logarithm of a number is logX= lnX / 2.303. Therefore, the natural logarithm of a number must be divided by the constant 2.303 to obtain the common logarithm of the same number

Order the following units from smallest to largest: Centimeter, angstrom, inch, mile, foot

Angstrom < Centimeter < Inch < Foot < Mile

When do you use the right hand rule? Explain the steps followed when using it?

It is used to determine the direction of the cross product resultant vector. 1. point thumb is the direction of vector A 2. extend fingers in the direction of vector B. You may need to rotate your wrist to get the correct configuration of thumb and fingers 3. Your palm establishes the plane between the two vectors. the direction your palm points os the direction of the resultant C

What are the units of Work and Energy (MKS) (SI)

Joule (J) (kg x M^2 / s^2)

How do you convert Celsius to Kelvin?

K = C + 273

Estimate log 7,426,135,420

Log between 7,426,135,420 about log (7.4 x 10^9) ~~ 9 + 0.74 = 9.74 (actual 9.87) even with this crazy number we can still accurate estimations by following basic logarithm rules

How do you convert Celsius to Fahrenheit?

Multiply the temperature by 1.8 and add 32.

What are the units of work in the (MKS) (SI)

Newton (N) = (kg x m / s^2)

What is the relationship between average velocity and average speed?

Average speed and average velocity may be unrelated because speed does not depend on displacement, but is rather the total distance traveled divided by the time.

what is the approximate pH of a solution with a pKa of 3.6, (HA)=100mM, and (A^-)= 0.1M? (Note: pH = pKa + log (A^-)/(HA)) A. 1.6 B. 3.6 C. 5.6 D. 7.6

B. This equation involves both a unit conversion between millimolar values and molar values, and calculation of a logarithm. The relationship between pH and pKa is described by the Henderson-Haselbalch equation given in the question stem. 100mM= 0.1M, so : pH= pKa + log (A^-)/(HA)=3.6 +log (1) = 3.6

using the appropriate number of significant digits, what is the answer to the following math problem? (Note: Assume all numbers are the result of measurements) 3.060 X 4.10 + 200. = A. 210 B. 213 C. 212.5 D. 212.55

B. While all digits are preserved during calculations, the final determination of the number of digits is made by both significant figures and decimal places. During multiplication, the answer is maintained to the smallest number of significant digits. During the addition, it is maintained to the smallest number of decimal places. By following the order of operations, addition is the last operation; thus we cannot have a decimal in our answer choice. Because multiplication occurred earlier, the result of that multiplication may be shortened according to the two significant figures in 4.10, but not the entire answer.

How would the number 17,060 be written in scientific notation? A. 1706x 10^1 B. 1.706 x 10^ 4 C. 1.7060 x 10^ 4 D. 0.17060 x 10^ 5

B. The question, while overtly testing the ability to use scientific notation, is also checking on the appropriate use of significant digits. Because there is no decimal point, the last zero is not significant and should not be used in scientific notation. The significand in scientific notation should always be between one and then.

A man walks 30 meters east and then 40 meters north. What is the different between his traveled distance and his displacement? A. 0m B. 20m C. 50m D. 70m

B. Using the pythagorean theorem, calculate the magnitude of the mans displacement: X =sqr (30^2 + 40^2) = 50 m His total distance traveled is equal to 30 + 40 = 70m. Therefore, the displacement between these two is 20m.

How does the number of significant digits differ between 14,320,010 and 3.618000? A. 14,320,010 has more significant digits than 3.61800 B. 14,320,010 has fewer significant digits than 3.61800 C. 14,320,010 has the same number of significant digits as 3.61800 D. A comparison cannot be made because the numbers are not both in scientific notation

C. Significant digits include all nonzero digits, all zeroes that are between nonzero digits, and trailing zeroes in any number with a decimal point. In 14,320,010 there is no decimal point; this the last zero is insignificant and there are seven digits. In 3.618000, all of the digits are significant; thus there are also seven significant digits.

A 150 pound man must be given a drug that is dosed at 1.5mg drug/ kg body mass. Approximately how many milligrams of the drug should be administered per dose? (Note: 1 lb = 4.45N) A. 33 mg B. 67 mg C. 100mg D. 225 mg

C. Because grams are a unit of mass and pounds are units of force, we must first convert pounds to newtons, and then divide by the acceleration due to gravity to find Kilograms. The weight of the person in Newtons is 150lb x 4.45N/lb = 667.5N. This corresponds to a mass of 667.5N/ 9m/s^s = 667.5 kg x m / s^2 / 9.81 m/ s^2 ~~ 67kg Now we can determine the dose: 67kg (1.5mg drug / kg body mass) ~~ 100mg 67 x 1.5 ~~ 100mg

which of the following expressions correctly illustrates the SI base units of the variables in the formula? m DeltaV = F delta t A. lb x mph = ft x lb x s B. kg x m/s = N x s C. kg x m/s = (kg x m / s^2) (s) D. g x m/s = (g x m / s^2) (s)

C. In SI units, mass is measured in kilograms (kg), velocity in meters per second (m/s), and the time in seconds (s). The newton is a derived unit, and is not considered to be a base unit of the Si system. A newton is equal to a N= (Kg x m / s^2)

At what given temperature do the Fahrenheit and Celsius scales give equal values? A. -40K B. 0K C. 233K D. 313K

C. This question requires not only unit conversions, but algebra as well. Given F= 9/5 C +32, the temperature T can calculate as: T= 9/5T + 32 --> 4/5T= 32 --> -5/4(32) = -40. However, the answers are given in Kelvin. -40C + 273 = 233K

Middle aged men require a base level of 900 calories per day plus an additional 12 calories per kilogram of body mass per day. Young adult women require a base level of 500 calories per day, plus 15 calories per kilogram of body mass per day. At what mass do middle-aged men and young adult women have the same caloric needs? A. 26kg B. 67kg C. 133kg D. 266kg

C. This is a system of equations couches in data. from this information we can construct two equations: C= 900 + 12m C= 500 + 15m These equations can be solved by setting them equal: 900 + 12m = 500 + 15m 400 = 3m 133kg = m

What are the magnitudes and the directions of the resultants from the following cross products? C = A x b and D = B x A A: X = -3 N, Y = 0 B: X = 0 , Y = +4 m

Te magnitude is simply the product of the magnitudes of the factor vectors and the sine of the angle between them. In this case, because one is oriented in the x-directions and the other is in the y-direction, the angle between them in 90 A x B x sin90 = 3N x4m x 1 = 12 N m the magnitude is therefore 12 N m To determine direction, for C, point right thumb toward left (negative x) and fingers point to the top of page (positive y) palm is into the page. Direction of D, thumb to right (Positive Y), fingers toward the left (negative x-direction) palm is pointing out of the page C is 12N m ( X (into the page) and D is 12 N m (out of the page)

How are the sine, cosine, and tangent calculated when given the dimensions of a right triangle? Sine: Cosine: Tangent:

The sine of the angle is equal to the ratio of the side opposite to the hypotenuse. SOH The Cosine is the ratio of the side adjacent to the angle to the hypotenuse. CAH Tangent is the ratio of the side opposite the angle to the side adjacent to the angle. TOA

Provide a definition for displacement or velocity in terms of the other variable.

velocity is the rate of the change in displacement of an object. Displacement is a function of velocity acting over a period of time.