In this multi-part exercise, we'll prove another result alluded to in class, namely that if the Pythagorean theorem holds in a neutral geometry, then the geometry is Euclidean. So, assume the Pythagorean theorem holds. Let ABC be an isosceles right triangle, with right angle at C and sides AC and BC equal. Let CM denote the altitude from C to the hypotenuse AB.
(a) Explain why M is the midpoint of AB.
(b) Prove that CM = AM (which, from part (a), is also equal to MB). Remember, we're assuming the Pythagorean theorem here.
(c) Use the result of part (b), and some angle-chasing, to prove that the sum of the angles of ABC is 180 degrees.
(d) Explain in one sentence why part (c) shows that the geometry is Euclidean.
In: Math
A Production Model predicts that the total output T is given in
terms of the amounts of labor x and capital investment y according
to the formula:
T = x p y q
for some constants p , q > 0 . Subject to a cost
constraint:
a x + b y = 1 ,
what x and y give the most total output T? What is the most
output?
In: Math
A manufacturer of tennis rackets finds that the total cost C(x) (in dollars) of manufacturing x rackets/day is given by
C(x) = 900 + 3x + 0.0003x2.
Each racket can be sold at a price of p dollars, where p is related to x by the demand equation
p = 5 − 0.0002x.
If all rackets that are manufactured can be sold, find the daily level of production that will yield a maximum profit for the manufacturer. Hint: The revenue is
R(x) = px,
and the profit is
P(x) = R(x) − C(x).
How Many Rackets?
[Hint:10,000 rackets is incorrect]
In: Math
5. Consider the function f(x) = -x^3 + 2x^2 + 2.
(a) Find the domain of the function and all its x and y intercepts.
(b) Is the function even or odd or neither?
(c) Find the critical points, all local extreme values of f, and the intervals on which f is increasing or decreasing.
(d) Find the intervals where f is concave up or concave down and all inflection points.
(e) Use the information you have found to sketch the graph of y= f(x).
In: Math
The volume of a sphere increases at 2.5 cubic cm/s. Calculate the rate at
which the area of the sphere changes when the diameter is 12 cm.
In: Math
(a) On the 24th day, the number of active cases are 53,697 and on the 26th day, the active cases increase to 82,272. Find the value of the constants in the above derived model.
The equation is: I(t)=K0ekt where k0 is the number of infected population present at t=0 and assuming the infected population grows continuously at a rate proportional to the number (of infected population)present.
(b) Predict the number of active cases on the 30th day.
(c) If we are to follow the model as derived in this problem, is there a point in time the active cases diminish?If yes, find the number of days it takes such that no person is infected. If not, state reasons.
(d) What are some of the steps you follow to decrease the spread of the virus (and making this world a safer place live in)?
In: Math
(1 point) Bacteria grow at a rate of 27% per hour in a petri dish. If there are initially 100 bacteria and a carrying capacity of 500000 cells, how long does it take to reach 94000 cells?
t = _____ hours
In: Math
Solve the following initial value problems
(1) dy/dt = t + y y(0) = 1 so y(t) =
(2) dy/dt = ty y(0) = 1 so y(t) =
In: Math
(A) Use logarithms to solve the problems.
How long will it take $15,000 to grow to $18,000 if the investment earns interest at the rate of 5%/year compounded monthly? (Round your answer to two decimal places.)
yr
How long will it take an investment of $5000 to triple if the investment earns interest at the rate of 3%/year compounded daily? (Round your answer to two decimal places.)
yr
Find the interest rate needed for an investment of $3000 to double in 5 years if interest is compounded continuously. (Round your answer to two decimal places.)
%/year
In: Math
Find the absolute maximum and minimum values for the function
f(x, y) = xy on the rectangle R defined by −8 ≤ x ≤ 8, −8 ≤ y ≤ 8.
In: Math
Compute the Taylor polynomial indicated. f(x) = cos(x), a = 0
T5(x) =
Use the error bound to find the maximum possible size of the error. Round your answer to nine decimal places.
cos(0.4) − T5(0.4) ≤
In: Math
Consider the following.
| optimize f(r, p) = 3r2 + rp − p2 + p |
| subject to g(r, p) = 3r + 4p = 1 |
(a) Write the Lagrange system of partial derivative equations. (Enter your answer as a comma-separated list of equations. Use λ to represent the Lagrange multiplier.)
(b) Locate the optimal point of the constrained system. (Enter
an exact number as an integer, fraction, or decimal.)
Once you have the answer matrix on the homescreen of your
calculator, hit MATH ENTER ENTER to convert any decimal
approximations to exact values. Do the same after you've evaluated
f at r and p to convert the approximated output value to an exact
value.
(r, p, f(r, p)) =
(c) Identify the optimal point as either a maximum point or a minimum point
In: Math
Find the volume of the solid obtained by rotating the region enclosed by the graphs of y=9−x, y=3x−3 and x=0 about the y-axis.
In: Math
A conical tank, point down, has a radius of 6 feet and a depth of 9 feet. Water is filling the tank and thus the volume of water in the tank is changing with time, as is the depth of the water and the radius of the surface of the water. Suppose that at the instant the depth of the water is 4 feet, the depth of the water is changing at an instantaneous rate of0.125 feet per minute. How fast is the volume of water in the tank changing at that instant?
In: Math
For each function, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal asymptote.
19. p(x)= 2x-3/x+4
21. s(x)=4/(x-2)^2
24. g(x)= 2x^2 +7x - 15/ 3x^2- 14+ 15
26. k(x)= 2x^2- 3x- 20/x-5
In: Math