What are the necessary an sufficient conditions for a sphere to be inscribed in a pyramid???
I can't believe nobody has answered your question. It is a beautiful question and I keep coming back to see if someone has answered it (not like "sv", of course). There has to be a relationship between the height and the radius of the circle which I haven't been able to find. I've drawn many times the circle and the triangle and I just can't get it!! Beautiful problem. I hope somebody solves it! -------------- Listen!. I asked a similar question so as to help in your quest and, interestingly, there is a formula I didn't know. Please check the answers to my question. That will lead you to your answer: http://answers.yahoo.com/question/index?...
How is right triangle geometry used in everyday life?
Right triangle geometry, and indeed trigonometry, and many other things the school is trying to teach you, are all used to help you graduate from college, and then to help you pass an interview so that you cam make $400.00 *everyday* at work instead of $52.40 *everyday*. I.e to make $100K a year instead of minimum wage. If the limit of your education is the knowledge that you already possess, you will learn nothing. Many of the other answers give specific examples for the use of trigonometry. What I'd like you to take away from this and share with others is that trigonometry, along with the rest of math and science, is a discovery and not an inventions. What you're learning is simply "how the world works". If that's not enough, and it should be, you're also learning how to learn. You're learning how to succeed at something that you didn't even understand when you started. You're learning how to make more of yourself than you knew possible. You're learning that your own self imposed limits are movable. You're learning that a skull full of mush can actually be molded into something useful. You're learning how to be more than you are now.
How can I prove Pythagoras theorem using my own legs?
Pythagorean Theorem is one of the most fundamental theorems in mathematics and it defines the relationship between the three sides of a right angled triangle. You are already aware of the definition and properties of a right angled triangle. It is the triangle with one of its angles as a right angle, that is, 90 degrees. The side that is opposite to the 90 degree angle is known as the hypotenuse. The other two sides that are adjacent to the right angle are called legs of the triangle.The theorem, also known as the Pythagorean theorem, states thatthe square of the length of the hypotenuse is equal to the sum of squares of the lengths of other two sides of the right-angled triangle. Or, the sum of the squares of the two legs of a right triangle is equal to the square of its hypotenuse.Let us call one of the legs on which the triangle rests as its base. The side opposite to the right angle is its hypotenuse, as we already know. The remaining side is called the perpendicular. So, mathematically, we represent the Pythagoras theorem as:Derivation:Consider a right angled triangle\Delta ABC. It is right angled at B.Let BD be perpendicular to the side AC.In \Delta ABC and \Delta ADBUsing the AA criterion for the similarity of triangles,ConsideringUsing the AA criterion for the similarity of triangles,Thus, it can be concludedSo if a perpendicular is drawn from the right-angled vertex of a right triangle to the hypotenuse, then the triangles formed on both sides of the perpendicular are similar to each other and also to the whole triangle.Now, we are required to prove . We drop a perpendicular BD on the side AC.We already know that∴ (Condition for similarity) Or …….. (1)Also, ∴ (Condition for similarity) Or ……….. (2)Summing equation (1) with equation (2),However,Thus,Hence, the proof of the Pythagoras theorem.Application of Pythagoras Theorem in Real Life:Pythagoras theorem is used to check if a given triangle is a right-angled triangle or not.Aerospace scientists and meteorologists find the range and sound source using the Pythagoras theorem.It is used by oceanographers to determine the speed of sound in water.
HELP!!!! CALCULUS 3 BONUS PROBLEMS!! ANSWER WHAT YOU CAN!!!!?
1.) Evaluate the line integral ∫_C vector F dot d =s, where vector F(x,y,z) = e^x i + e^x-y j + e^y k, and C is the path consisting of straight line segments form (0,0,0) to (0,0,1) and then from (0,0,1) to (0,1,1). 2.) Consider the sphere x^2+ y^2 +z^2 = 4a^2 of radius 2a and center at the origin. use a surface integral to find a formula for the area of the cap, S, of the sphere above the disk x^2 + y^2 = a^2 in the xy- plane.( hint: use polar coordinates). 3.) Suppose ∫ upside down triangle with a weird looking symbol that kind of looks like ℓ, but more curvy dot d (vector)s = 1 where T is the path from P = (-2,1) to q = (2,1) along the top[ half of the ellipse 3x^2 + 11y^2 = 23. Determine the value of ∫upside down triangle with a weird looking symbol that kind of looks like ℓ, but more curvy dot d (vector)s where B is the path from P to Q along the bottom half of that ellipse. 4.) Find the flow rate of water with velocity field vector v(x,y,z) = (2x,y,xy) meters per second across the part of the cylinder x^2 + y^2 = 9 with x,y ≥ 0, and 0 ≤ z ≤ 4.). (x,y,z measured in meters). 5.) Use Greens Theorem to compute the are above the x-axis and under one arch of the cycloid given parametrically by x = f(t) = t - sint, y = g(t) = 1- cost, 0 ≤ t ≤ 2π.
Name of mathmatican and thier achievments also?
1) Pythagoras of Samos (ca 578-505 BC) Greece - Pythagoras believed that all relations could be reduced to number relations The Pythagorean Theorem - The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.( a² + b² = c²) Pythagoras of Samos http://www.gap-system.org/~history/Mathe... 2) Carl Gauss Carl Gauss - (1777-1855) Germany - Gauss built the theory of complex numbers into its modern form - wrote the first modern book on number theory, and proved the law of quadratic reciprocity Karl Friedrich Gauss - Child Mathematical Genius http://math.about.com/cs/mathematicians/... 3) Isaac Newton (1642-1727) England - made contributions to all branches of mathematics - famous for his solutions in analytical geometry ( drawing tangents to curves etc) Isaac Newton Institute for Mathematical Sciences http://www.newton.ac.uk/newtlife.html Here is a very good site that lists 60 mathematicians - take your pick!! The Greatest Mathematicians of All Time http://fabpedigree.com/james/mathmen.htm
CALCULUS HOMEWORK HELP PLEASE!?
1) f(x) = bx^2 +1 { x< (or equal to ) 2 = ax-3 { x> 2 Let f be the function above. If f(x) is differentiable at x2, which of the following is the value of a?--> a) 5 b) 4 c) 3 d) 2 e)1 2) Let g be a twice differentiable function with g'(x)>0 and g''(x0< 0 for all real numbers x such that g(5)=-15 and g(6)= - . Of the following which is the possible value of g(7)?--> a) -11 b)-7 c)0 d)1 e)2 3) A particle moves along x-axis such that any time t>(or equal to ) 0, its velocity is given v(t)=7+2tsin(.33t). What is the acceleration of the particle at t=3? 4) The volume of a sphere is increasing at a constant rate of 4pie cubic meters per second. What is the rate of increase in the radius of the sphere at the instant when the surface area of the sphere is 16pie square meters? 5) A cylindrical water tank that is filled to a capacity of 500 gallons is leaking at a rate of -60e^0.1t gallons per our. TO the nearest hour, what is the time elapsed when the volume of water remaining in the tank is 10 gallons? 6) What is the integral of 3xradical(x+2)dx? 7) A window in the shape of a rectangle capped with an equilateral triangle is to have an area of 125 square yards. What is the minimum perimeter of such a window? 8) Let g be the function given by g(x)= integral from 0 to 5x^2 of sin (2t)dt. What is g'(x)?