Inna Shpiro, M.A.
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About
I am an Adjunct Math and Physics instructor and professional Math/Physics tutor.
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Recent Activity

Answered a Question in Math
Let's consider what information we have available about the quantities in the problem. The total cost of cows, or the income available from the sale of cows, is a perfect square. It is implied... 
Answered a Question in Math
Denote the center of the circle with the diameter AB as `O_1` and the center of the circle with the diameter CD as `O_2` ` ` . (Please see the attached picture for reference.) Then, the triangles... 
Answered a Question in Math
The position vector is `r(t) = <t, t^2  1>` a) The components of the velocity vector are the derivatives of the components of the position vector: `v(t) = <1, 2t>` The speed is the... 
Answered a Question in Math
I am assuming that 3 after the first log in both expressions is supposed to be the base of the logarithm. (The second 3 can be either a base of the log or a coefficient of x.) Then, the task is to... 
Answered a Question in Science
The reason for the interpreting Newton's Second Law as focused on solving for acceleration rather then net force is that the acceleration determines the rate of change of velocity of an object. The... 
Answered a Question in Math
To find the minimum value of the given expression, let's first rewrite is as a function of x, given the constraint 20x + 21y = 1247. From here, we can find y: `y = (1247  20x)/21` and substitute... 
Answered a Question in Math
Face value, when referring to a loan, means the amount borrowed. This is the amount written on the document detailing the terms of the loan—hence the term "face value." Since the loan will accrue... 
Answered a Question in Math
This question is about ratios and proportions. A large object, such as a tower, has dimensions that are measured in feet. A small model of this object has dimensions that are measured in inches.... 
Answered a Question in Science
The entropy change for a process is determined by `dS = (dQ)/T` , where `dQ ` is the heat transfer, and T is the temperature. By definition, the temperature stays constant during an isothermal... 
Answered a Question in Science
The magnetic field lines of a long currentcarrying wire are concentric circles in the plane perpendicular to the wire. The direction of the field is determined by the righthand rule. If the wire... 
Answered a Question in Science
The work required to bring the charges to the given points equals the electric potential energy of the system of two charges at these points. Since both charges are negative, the electrostatic... 
Answered a Question in Science
This problems presents a symmetric (cylindrical) charge distribution. The electric field of such distribution can be found using Gauss's Law, which states that the electric flux through a given... 
Answered a Question in Math
Angular momentum, along with other physical quantities, such as linear momentum and energy, is a conserved quantity. The equation describing the change of angular momentum is similar to that of... 
Answered a Question in Math
The motion of the car can be broken up into two parts: in the first part, the car is moving with constant velocity, and in the second part, the car is decelerating. The distance traveled has to be... 
Answered a Question in Math
The average velocity over a time interval, by definition, is the displacement that happened during that time interval, divided by the time interval: `vec v_(avg) = (vec Delta r)/(Delta t)` The... 
Answered a Question in Math
The relationship between velocities in different reference frames is `vecv_(mo) = vecv_(mw)+vecv_(wo)` Here, the subscript m stands for "man," w for "woman," and o for stationary observer. This... 
Answered a Question in Math
The point of tangency of the graphs of two curves is a point where they touch but do not intersect each other. This happens when the curves have a common tangent line at this point. (Recall that... 
Answered a Question in Math
We have to prove that there exists no integer numbers a and b such that 18a = 6b + 1. For 6b + 1 to equal 18 times an integer number, 6b + 1 has to be a multiple of 18. If an integer is a multiple... 
Answered a Question in Math
Linear momentum is a vector quantity describing the motion of an object or a system of objects. In the past, it was also referred to as a "quantity of motion." The mathematical expression for the... 
Answered a Question in Science
Technically, the answer is to the first question is "neither." An atom, by definition, is a fundamental unit of matter, and it is neutral: it has a positively charged nucleus and several negatively... 
Answered a Question in Math
We have to calculate the following quantity: `log_(1/8) sin(4350)` , where 4350 is the angle in degrees. First, let's simplify the sine of the angle, and, possibly, determine its value. To do that,... 
Answered a Question in Math
Let's translate all the information given in the problem to algebraic expressions. The girl's age at present time is denoted by the variable g. Also, at present, the girl's father is eleven times... 
Answered a Question in Math
We are given two functions: `f(x) = 5^x` and `g(x) = sqrt(x)` a) In i), we have to determine f(2), that is, the value of the function f(x) when x = 2. This can be done simply by plugging in x = 2... 
Answered a Question in Math
Multivariable equations are typically known as linear equations or formulas. Formulas are commonly used in sciences, geometry, business, and other fields to indicate the connection between several... 
Answered a Question in Math
A relation is a rule that defines a correspondence between two numbers. The first number is selected from the "input" set, also called domain, and the second number is selected from the "output"... 
Answered a Question in Math
A general quadratic equation in the form `ax^2 +bx + c = 0` can have two real roots, one real root, or two complex roots. To recognize an equation that has complex roots (without solving it), one... 
Answered a Question in Math
Direct variation is a relationship in which one variable is proportional to another. For independent variable x and dependent variable y, the general direct variation equation looks as follows: `y... 
Answered a Question in Math
Generally, cubic binomials are algebraic expressions consisting of two terms, one of which has a variable taken to the third power ("cubed"). However, the typical cubic binomial you will have to... 
Answered a Question in Math
Compound inequalities are combinations of two or more inequalities. There are two different ways to combine, or "compound", inequalities: by using either "AND" or "OR". "AND" means the two... 
Answered a Question in Science
Electric field, by definition, is a force on a unit electric charge. Electric fields are typically represented by lines with arrows. At any given point on the line, the direction of the field is... 
Answered a Question in Math
The general equation of the pointslope form is `y  y_0 = m(x  x_0)` Sometimes you might also see it written as `y  y_1 = m(x  x_1)` . Here, x and y are independent and dependent variables,... 
Answered a Question in Math
The easiest way to graph the function given by the equation `y = x^2` is by creating a table of ordered pairs. This means picking a value of x, calculating y (by squaring x), and graphing the... 
Answered a Question in Math
One cannot really "solve" a function; the appropriate term is to evaluate a function. A piecewise function is typically given by different formulas on different intervals of independent variable,... 
Answered a Question in Math
This question is a little too broad, and it would be easier to discuss if the situation in which you need to write a piecewise function was known: for example, if you are modeling a reallife... 
Answered a Question in Math
The best way to illustrate how to solve equations with variables on both side is by considering an example. Let's look at a linear equation with variables on both sides but no other complications... 
Answered a Question in Math
The general formula for the average value of the function `f(x) ` on the given interval [a, b] is `1/(ba) int_a ^b f(x)dx` In this case, the given function is `f(t) = e^(4t)(cos6t + 3sin6t)` and... 
Answered a Question in Math
Finding antiderivatives is a process that, as the name suggests, is the reverse of that of taking a derivative. It is the same thing as finding an indefinite integral of a function. The common... 
Answered a Question in Reference
To solve the equation 5sinx = 4cosx, first, bring all terms to one side of the equations, so that the other side becomes 0: 5sinx  4cosx = 0. Then, factor out 4cosx from the right hand side:... 
Answered a Question in Reference
The multiplicative inverse of an element of a set is another element of this set such that the product of the two elements is a multiplicative identity. In the given set of complex numbers, the... 
Answered a Question in Science
This question can be answered using Boyle's Law. Notice that initially the oxygen gas and the helium gas have the same temperature, and when they are both pumped into a tank, they remain at the... 
Answered a Question in Reference
In this question, I am not clear whether the logs have the base of 3 or 3 is the part of the arguments. I will try to show both cases. Let's consider the given system of equations: 4^(x/y)*4^(y/x)... 
Answered a Question in Math
In order to write the equation of a line, we need to know either two points that belong to that line or one point and the line's slope. In this case, since the line "hits the origin" at a 45degree... 
Answered a Question in Reference
The multiplicative inverse is another name for the reciprocal. Both numbers are such that, if multiplied by the given number, they result in 1. The number 1 is special with regard to the operation... 
Answered a Question in Math
I am assuming that 3 here is supposed to be an exponent of u and that the dot is supposed to represent multiplication, so the question should be to simplify `u^3*u^3*u` . Then, use the rules of... 
Answered a Question in Reference
The notation of a circle between two functions (for example, f o g (x)) means the composition of two functions. It is found as function f (the one that comes first) of g(x): f o g (x)= f(g(x)). In... 
Answered a Question in Science
A twodimensional vector is a geometrical object that can be given either as a magnitude and direction, or as two components. When the two components are given, as in this problem, the magnitude... 
Answered a Question in Nickel and Dimed: On (Not) Getting by in America
In addition to the examples of the problems experienced by Holly and Lucy, Nickel and Dimed illustrates vividly the importance of access to healthcare when Barbara herself has a medical issue.... 
Answered a Question in Science
The correct answer choice is d): the statement that atoms are like sold little balls best reflects Dalton's thinking at the time. Dalton proposed that atoms are indivisible and indestructible... 
Answered a Question in trigonometry math
Since this trigonometric sum involves the angles that are multiples of pi/3 (equivalent to 60 degrees)—a "special" angle with known sine, cosine, and tangent values—each of the terms can be... 
Answered a Question in Reference
In the situation described in this problem, the hammer can be modeled as an object undergoing uniform circular motion. In this kind of motion, the speed of the object is constant, but since the...
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