An easy way to do this one is to use a graphical calculator (e.g. TI-84) to plot the function inside the square root. The calculator will show that 2 x/(x2 - 324) is defined and non negative in two intervals: x > 18 and -18 < x ≤0 The...
An easy way to do this one is to use a graphical calculator (e.g. TI-84) to plot the function inside the square root. The calculator will show that 2 x/(x2 - 324) is defined and non negative in two intervals: x > 18 and -18 < x ≤0 The...
The original equation can be solved using the quadratic formula and removing a factor of i from the square root that appears in the quadratic formula. The result is x = cosy ± i siny. By the Euler theorem this is x = ...
The equation of a plane in 3d space can be written as α x + β y + γ z = 1 for some value of the parameters α, β, γ The fact that point A is in this plane means that α - 2 β + 6 γ =1 ...
The parabola y = 9 x2 +7 x +7 is in standard form with a = 9, b = 7 and c =7. The x coordinate of the vertex is give by -b/(2a). In this case that works out to be -7/18.
Think of j , k and p as vectors and introduce a scalar called λ. Then a general point on the line joining the points j and k can be written as the vector equation p = j + λ (k - j) (so when λ = 0 we have p = j and when...
It is easy to guess how this works out. That is the angles x,y,z must be all be at mutual angles of 120 degrees. For example x = 0, y = 120, z = 240. Then cos(y-z) + cos(z-x) + cos(x-y) does equal -3/2 and cosx + cosy...
My first observation about this problem is that a tire diameter of 8.0m is unrealistically large. Perhaps on a giant mining machine - but not on a car. Another issue is the non SI unit of km/hr. This must be changed to m/s. 80 km/hr = 22.2 m/s and 30 km/hr...
I am not sure what your specific question is, but perhaps the answer is "all of the above". All four statements are true about the function f(x). The small exception is that the function has a point of inflection at x = 0, but the value of the function is not zero at that point...
You are dealing with an infinite product. Call the infinite product Q. Convert the infinite product to an infinite sum by taking the log2 of both sides. This results in: log2(Q) = ∑ n=1∞ n/5n This...
Since you have the diagram, you will have noticed that the triangle formed by the the two walks is a right triangle with legs of 7 and 10 km. This means that the hypotenuse of the triangle is h = sqrt(49 +100). The next step is to notice that the angle formed by the...
There are two possible answers to this question. If the dice are considered distinguishable (that is a 2 on the red die and a 5 on the green die is considered a different outcome than a 5 on the red die and a 2 on the green one) then there are 36 (= 6 x 6) possible outcomes. However,...
Assuming that the the 5 m travel mentioned is in the last TWO seconds of its travel, and that the end of travel means the the part just before reaching the maximum height, the answer is that there is no difference between the ball thrown at 100 m/s and the one one thrown at 200 m/s. A...
From the information given, one may conclude that four of the seven numbers are 2,3,3,20. (the number 3 is the only number appearing twice. It can appear only twice if the median is 6) The only way that the median can be 6 is if the number...
To find the extremum, the derivative of f(x) = x ln(2x) must be calculated and set to zero. The derivative f' can be found with the aid of the chain rule. It is f' = 1 + ln(2) + ln(x) Setting this...
Since C1 and C2 are in parallel they add to result in C1 + C2. This combination ins in series with C3 so 1/Ceq = 1/(C1 + C2) + 1/C3. The values can be plugged in to get 1/Ceq = .2165 x 10^6 ...
The key equation is the lens equation: 1/p + 1/q = 1/f where p is the object distance, q is the image distance and f is focal length. Since the image distance is equal to the object distance we can write 2/p = ...
This problem can be solved by writing S20 / S10 = 2 a2 / a1 = 2( a1 + d )/ a1 Use the formula for the sum of the first n terms of an arithmetic series: Sn = n (2 a1 +(n-1) d )/2 where a1 is the first...
The set of integers is a countable infinite set. That is, the integers can be arrange in ordinal number (first, second, third, ...) order. For example {0, 1, -1, 2, -2, 3 , -3, 4, ...}. The real numbers cannot be so arranged. In set theory this means that the set...
The basic physics formula is V/V0 = exp(-(t-t0)/RC)) where RC is the RC time constant Taking the ln of both sides and rearranging gives t-to = RC ln(V0/V) = RC ln(RV0/RV) ...
The ratio test shows that this series will converge for abs(x/4) <1. The boundary cases x = 4 and x = -4 need to be checked. For x = 4, the series is the harmonic series which does not converge. For x = -4, the series is an alternating sign series...