"If you are wearing a hat, you are a baseball fan." and "If you are not a baseball fan, you are not wearing a hat." It's called a [B]contrapositive[/B]. definition: : a proposition or theorem formed by contradicting both the subject and predicate or both the hypothesis and conclusion of a given proposition

A 13ft ladder leans against the side of a house. The bottom of the ladder is 10ft from the side of the house. How high is the top of the ladder from the ground? If necessary, round your answer to the nearest tenth. We have a right triangle. Hypotenuse = 13, one leg = 10. We use our [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=10&hypinput=13&pl=Solve+Missing+Side']Pythagorean theorem Calculator to solve for the other leg[/URL]: s = [B]8.3066[/B]

A 50-foot pole and a 70-foot pole are 30 feet apart. If you were to run a line between the tops of the two poles, what is the minimum length of cord you would need? The difference between the 70 foot and 50 foot pole is: 70 - 50 = 20 foot height difference. So we have a right triangle, with a height of 20, base of 30. We want to know the hypotenuse. Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=20&side2input=30&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator to solve for hypotenuse[/URL], we get: hypotenuse = [B]36.06 feet[/B]

A certain group of woman has a 0.69% rate of red/green color blindness. If a woman is randomly selected, what is the probability that she does not have red/green color blindness? 0.69% = 0.0069. There exists a statistics theorem for an event A that states: P(A) + P(A') = 1 where A' is the event not happening In this case, A is the woman having red/green color blindness. So A' is the woman [U][B][I]not[/I][/B][/U][I] having red/green color blindness[/I] So we have: 0.0069 + P(A') = 1 Subtract 0.0069 from each side, we get: P(A') = 1 - 0.0069 P(A') = [B]0.9931[/B]

A ladder 25 feet long is leaning against a wall. If the base of the ladder is 7 feet from the wall, how high up the wall does the ladder reach? We have a right triangle, where the ladder is the hypotenuse, and we want the measurement of one leg. Set up the pythagorean theorem with these given items using our P[URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=7&hypinput=25&pl=Solve+Missing+Side']ythagorean Theorem Calculator[/URL]. We get Side 1 = [B]24 feet.[/B]

A ladder is 25 ft long. The ladder needs to reach to a window that is 24 ft above the ground. How far away from the building should the bottom of the ladder be placed? We have a right triangle, where the ladder is the hypotenuse, and the window side is one side. Using our right triangle and the [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=24&hypinput=25&pl=Solve+Missing+Side']pythagorean theorem calculator[/URL], we get a length of [B]7 ft [/B]for the ladder bottom from the wall.

A young dad, who was a star football player in college, set up a miniature football field for his five-year-old young daughter, who was already displaying an unusual talent for place-kicking. At each end of the mini-field, he set up goal posts so she could practice kicking extra points and field goals. He was very careful to ensure the goalposts were each straight up and down and that the crossbars were level. On each set, the crossbar was six feet long, and a string from the top of each goalpost to the midpoint between them on the ground measured five feet. How tall were the goalposts? How do you know this to be true? The center of each crossbar is 3 feet from each goalpost. We get this by taking half of 6, since midpoint means halfway. Imagine a third post midway between the two goal posts. It has the same height as the two goalposts. From the center post, the string from the top of a goalpost to the base of the center post, and half the crossbar form and right triangle with hypotenuse 5 feet and one leg 3 feet. [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=3&hypinput=5&pl=Solve+Missing+Side']Using the Pythagorean Theorem[/URL], the other leg -- the height of each post -- is 4 feet.

Free Algebra Master (Polynomials) Calculator - Given 2 polynomials this does the following:
1) Polynomial Addition
2) Polynomial Subtraction

Also generates binomial theorem expansions and polynomial expansions with or without an outside constant multiplier.

Free Chebyshevs Theorem Calculator - Using Chebyshevs Theorem, this calculates the following:
Probability that random variable X is within k standard deviations of the mean.
How many k standard deviations within the mean given a P(X) value.

CHEBYSHEVS THEOREM TELLS US THAT WHAT PERCENTAGE LIES BETWEEN 2.25 STANDARD DEVIATIONS? Using our [URL='http://www.mathcelebrity.com/chebyshev.php?pl=probability&k=2.25&probk=0.75']Chebyshevs Theorem calculator[/URL], we get: P(X - u| < kσ) >= [B]0.802469[/B]

Free Chinese Remainder Theorem Calculator - Given a set of modulo equations in the form:
x ≡ a mod b
x ≡ c mod d
x ≡ e mod f

the calculator will use the Chinese Remainder Theorem to find the lowest possible solution for x in each modulus equation.
Given that the n_i portions are not pairwise coprime and you entered two modulo equations, then the calculator will attempt to solve using the Method of Successive Subsitution

Compute a 75% Chebyshev interval around the mean for [I]x[/I] values and also for [I]y[/I] values. [B][U]Grid E: [I]x[/I] variable[/U][/B] 11.92 34.86 26.72 24.50 38.93 8.59 29.31 23.39 24.13 30.05 21.54 35.97 7.48 35.97 [B][U]Grid H: [I]y[/I] variable[/U][/B] 27.86 13.29 33.03 44.31 16.58 42.43 39.61 25.51 39.14 16.58 47.13 14.70 57.47 34.44 According to Chebyshev's Theorem, [1 - (1/k^2)] proportion of values will fall between Mean +/- (k*SD) k in this case equal to z z = (X-Mean)/SD X = Mean + (z*SD) 1 - 1/k^2 = 0.75 - 1/k^2 = 0.75 - 1= - 0.25 1/k^2 = 0.25 k^2 = 1/0.25 k^2 = 4 k = 2 Therefore, z = k = 2 First, [URL='http://www.mathcelebrity.com/statbasic.php?num1=11.92%2C34.86%2C26.72%2C24.50%2C38.93%2C8.59%2C29.31%2C23.39%2C24.13%2C30.05%2C21.54%2C35.97%2C7.48%2C35.97&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']determine the mean and standard deviation of x[/URL] Mean(x) = 25.24 SD(x) = 9.7873 Required Interval for x is: Mean - (z * SD) < X < Mean + (z * SD) 25.24 - (2 * 9.7873) < X < 25.24 - (2 * 9.7873) 25.24 - 19.5746 < X < 25.24 + 19.5746 5.6654 < X < 44.8146 Next, [URL='http://www.mathcelebrity.com/statbasic.php?num1=27.86%2C13.29%2C33.03%2C44.31%2C16.58%2C42.43%2C39.61%2C25.51%2C39.14%2C16.58%2C47.13%2C14.70%2C57.47%2C34.44&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']determine the mean and standard deviation of y[/URL] Mean(y) = 32.29 SD(y) = 9.7873 Required Interval for y is: Mean - (z * SD) < Y < Mean + (z * SD) 32.29 - (2 * 13.1932) < Y < 32.29 - (2 * 13.1932) 32.29 - 26.3864 < Y < 32.29 + 26.3864 5.9036 < X < 58.6764

Free Cubic Equation Calculator - Solves for cubic equations in the form ax³ + bx² + cx + d = 0 using the following methods:
1) Solve the long way for all 3 roots and the discriminant Δ
2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.

Free Demoivres Theorem Calculator - Using Demoivres Theorem, this calculator performs the following:
1) Evaluates (acis(θ))ⁿ
2) Converts a + bi into Polar form
3) Converts Polar form to Rectangular (Standard) Form

Free Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:
* Difference of Squares
* Sum of Cubes
* Difference of Cubes
* Binomial Expansions
* Quadratics
* Factor by Grouping
* Common Term
This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots
* Factors and simplifies Rational Expressions of one fraction
* Determines the number of potential positive and negative roots using Descarte’s Rule of Signs

Free Fermats Little Theorem Calculator - For any integer a and a prime number p, this demonstrates Fermats Little Theorem.

Gigi’s family left their house and drove 14 miles south to a gas station and then 48 miles east to a water park. How much shorter would their trip to the water park have been if they hadn’t stopped at the gas station and had driven along the diagonal path instead? [IMG]https://mathcelebrity.com/community/data/attachments/0/pythag-diagonal.jpg[/IMG] Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=14&side2input=48&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we see the diagonal route would be: 50 miles The original trip distance was: Original Trip Distance = 14 + 48 Original Trip Distance = 62 miles Diagonal Trip was 50 miles, so the difference is: Difference = Original Trip Distance - Diagonal Distance Difference = 62 - 50 Difference = [B]12 miles[/B]

Given the rectangular prism below, if AB = 6 in., AD = 8 in. and BF = 24, find the length of FD. [IMG]http://www.mathcelebrity.com/images/math_problem_library_129.png[/IMG] If AB = 6 and AD = 8, by the Pythagorean theorem, we have BD = 10 from our [URL='http://www.mathcelebrity.com/pythag.php?side1input=6&side2input=8&hypinput=&pl=Solve+Missing+Side']Pythagorean Theorem[/URL] Calculator Using that, we have another right triangle which we can use the [URL='http://www.mathcelebrity.com/pythag.php?side1input=10&side2input=24&hypinput=&pl=Solve+Missing+Side']pythagorean theorem[/URL] calculator to get [B]FD = 26[/B]

Free Lagrange Four Square Theorem (Bachet Conjecture) Calculator - Builds the Lagrange Theorem Notation (Bachet Conjecture) for any natural number using the Sum of four squares.

Free Limit of a Function Calculator - This lesson walks you through what limit is, how to write limit notation, and limit theorems

Free Normal Distribution Calculator - Calculates the probability that a random variable is less than or greater than a value or between 2 values using the Normal Distribution z-score (z value) method (Central Limit Theorem).
Also calculates the Range of values for the 68-95-99.7 rule, or three-sigma rule, or empirical rule. Calculates z score probability

Free Pick's Theorem Calculator - This calculator determines the area of a simple polygon using interior points and boundary points using Pick's Theorem

Free Pythagorean Theorem Calculator - Figures out based on user entry the missing side or missing hypotenuse of a right triangle. In addition, the calculator shows the proof of the Pythagorean Theorem and then determines by numerical evaluation if the 2 sides and hypotenuse you entered are a right triangle using the Pythagorean Theorem

Free Pythagorean Theorem Trig Proofs Calculator - Shows the proof of 3 pythagorean theorem related identities using the angle θ:
Sin²(θ) + Cos²(θ) = 1
Tan²(θ) + 1 = Sec²(θ)
Sin(θ)/Cos(θ) = Tan(θ)

Free Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax² + bx + c = 0. Also generates practice problems as well as hints for each problem.
* Solve using the quadratic formula and the discriminant Δ
* Complete the Square for the Quadratic
* Factor the Quadratic
* Y-Intercept
* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)² + k
* Concavity of the parabola formed by the quadratic
* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.

Free Quartic Equations Calculator - Solves quartic equations in the form ax⁴ + bx³ + cx² + dx + e using the following methods:
1) Solve the long way for all roots and the discriminant Δ
2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.

Free Quotient-Remainder Theorem Calculator - Given 2 positive integers n and d, this displays the quotient remainder theorem.

Free Right Triangles Calculator - This solves for all the pieces of a right triangle based on given inputs using items like the sin ratio, cosine ratio, tangent ratio, and the Pythagorean Theorem as well as the inradius.

Running from the top of a flagpole to a hook in the ground there is a rope that is 9 meters long. If the hook is 4 meters from the base of the flagpole, how tall is the flagpole? We have a right triangle, with hypotenuse of 9 and side of 4. [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=4&hypinput=9&pl=Solve+Missing+Side']Using our Pythagorean Theorem calculator[/URL], we get a flagpole height of [B]8.063[/B].

Sam leaves school to go home. He walks 10 blocks North and then 8 blocks west. How far is John from the school? Sam walked at a right angle. His distance from home to school is the hypotenuse. Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=8&side2input=10&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we get: [B]12.806 blocks[/B]

We know from the pythagorean theorem: [SIZE=5][B]sin^2(x) + cos^2(x) = 1[/B] [B]Subtract sin^2(x) from each side and we get:[/B] [B][B]cos^2(x) = 1 - [B]sin^2(x)[/B][/B][/B] [B][B][B]We can rewrite our original expression as:[/B][/B][/B] [B][B][B]sin^2(x)/cos^2(x)[/B][/B][/B] [/SIZE] [B][B][B][SIZE=5]But this expression [/SIZE][SIZE=4]equals[/SIZE][SIZE=5] tan^2(x)[/SIZE][/B][/B][/B] [MEDIA=youtube]zqYg0VRq5Ak[/MEDIA]

Suppose x is a natural number. When you divide x by 7 you get a quotient of q and a remainder of 6. When you divide x by 11 you get the same quotient but a remainder of 2. Find x. [U]Use the quotient remainder theorem[/U] A = B * Q + R where 0 ≤ R < B where R is the remainder when you divide A by B Plugging in our numbers for Equation 1 we have: [LIST] [*]A = x [*]B = 7 [*]Q = q [*]R = 6 [*]x = 7 * q + 6 [/LIST] Plugging in our numbers for Equation 2 we have: [LIST] [*]A = x [*]B = 11 [*]Q = q [*]R = 2 [*]x = 11 * q + 2 [/LIST] Set both x values equal to each other: 7q + 6 = 11q + 2 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=7q%2B6%3D11q%2B2&pl=Solve']equation calculator[/URL], we get: q = 1 Plug q = 1 into the first quotient remainder theorem equation, and we get: x = 7(1) + 6 x = 7 + 6 [B]x = 13[/B] Plug q = 1 into the second quotient remainder theorem equation, and we get: x = 11(1) + 2 x = 11 + 2 [B]x = 13[/B]

Free Synthetic Division Calculator - Using Ruffinis Rule, this performs synthetic division by dividing a polynomial with a maximum degree of 6 by a term (x ± c) where c is a constant root using the factor theorem. The calculator returns a quotient answer that includes a remainder if applicable. Also known as the Rational Zero Theorem

Draw this rectangle and you'll see we have a pythagorean theorem equation. a^2 + b^2= c^2 b = 8 and c= 10 a^2 + 8^2 = 10^2 a^2 + 64 = 100 Subtract 64 from each side: a^2 = 36 a= 6 Therefore, perimeter P is: P = 2l + 2w P = 2(6) + 2(8) P = 12 + 16 P = [B]28[/B] [MEDIA=youtube]8lcpRet3r18[/MEDIA]

The distance between consecutive bases is 90 feet. An outfielder catches the ball on the third base line about 40 feet behind third base. How far would the outfielder have to throw the ball to first base? We have a right triangle. From home base to third base is 90 feet. We add another 40 feet to the outfielder behind third base to get: 90 + 40 = 130 The distance from home to first is 90 feet. Our hypotenuse is the distance from the outfielder to first base. [URL='https://www.mathcelebrity.com/pythag.php?side1input=130&side2input=90&hypinput=&pl=Solve+Missing+Side']Using our Pythagorean theorem calculator[/URL], we get: d = [B]158.11 feet[/B]

The height of an object t seconds after it is dropped from a height of 300 meters is s(t)=-4.9t^2 +300. Find the average velocity of the object during the first 3 seconds? (b) Use the Mean value Theorem to verify that at some time during the first 3 seconds of the fall the instantaneous velocity equals the average velocity. Find that time. Average Velocity: [ f(3) - f(0) ] / ( 3 - 0 ) Calculate f(3): f(3) = -4.9(3^2) + 300 f(3) = -4.9(9) + 300 f(3) = -44.1 + 300 f(3) = 255.9 Calculate f(0): f(0) = -4.9(0^2) + 300 f(0) = 0 + 300 f(0) = 300 So we have average velocity: Average velocity = (255.9 - 300)/(3 - 0) Average velocity = -44.1/3 Average velocity = -[B]14.7 [/B] Velocity is the first derivative of position s(t)=-4.9t^2 +300 s'(t) = -9.8t So we set velocity equal to average velocity: -9.8t = -14.7 Divide each side by -9.8 to solve for t, we get [B]t = 1.5[/B]

The triangle sum theorem states the sum of the three angles in a triangle equals 180 degrees. So if you're given two angles and need too find the 3rd angle, add the 2 known angles up, and subtract them from 180 to get the 3rd angle measure.