Chinenye O.

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Trigonometry

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Question:

The area of a right triangle is 60. One of its angles is 45 degrees. Find the length of the sides and the hypotenuse of the triangle.

Chinenye O.

Answer:

The solution will be best with a sketch of the right triangle but since this platform could not accommodate that, let's imagine it: Let's visualize the angles; - Since it is a right triangle, it has one of the angles as 90 degrees. - And another angle is given as 45 degrees. - Therefore, the third angle will be 180 - [90 + 45] = 45 degree............ (Sum of the angles of a triangle) - So we have the angles as 90 degrees, 45 degrees and 45 degrees.This implies that we have an isosceles right triangle. We are given the area as 60. And the area of a triangle = 1/2(base x height). Let the base be b and the height is a. S we have: 1/2ab = 60 ........................................................................Equation 1 Notice that the sides of the triangle correspond to the base and the height. And since the triangle is isosceles, then the base and the height are equal. So, we can safely rewrite equation 1 as: 1/2( b x b) = 60 .......................................................................................... Equation 2 (where b denotes the base, which is also equal to the height) 1/2(2b) = 60 Therefore, b = 60. So, we conclude that the length of the sides is equal to 60. To find the hypotenuse, we use the Pythagorean's theorem: (The square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides) c^2 = a^2 + b^2 .............................................................................................. Equation 3 c^2 = (60)^2 + (60)^2 c^2 = 3600 + 3600 c^2 = 7200 c = Square root of 7200 = 84.85 (to 2 decimal places). Summary: - Side a = 60 - Side b = 60 - Hypotenuse = 84.85

Pre-Algebra

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Question:

The area of the circular base of a cylinder is 4π. The height of the cylinder is 7. Find the volume of the cylinder.

Chinenye O.

Answer:

Solving this problem requires you visualizing the solid shape. So, let's go with the figure below: *** This platform was unable to accommodate the intended figure. *** So, let's use our imagination: The formula for finding the volume of a cylinder is πhr^2 (This formula is easy to prove but let's just work with it for now). Volume of a cylinder = πhr^2 ------------------------------------------------Equation 1 We know the following: - h = 7 - r is unknown So, we need to find the value of r in order to evaluate equation 1. But the formula for finding the area of the circular base of a cylinder is πr^2. So let's equate this to the value given in the problem: πr^2 = 4π -----------------------------------------------------------------------------Equation 2 So, let's solve for r in equation 2: - Divide both sides by π: r^2 = 4 - Take the square root of both sides: r = 2. Now, let's substitute the values of r and h in equation 1: Volume of the cylinder = πhr^2 = π[7(2^2)] = 28π

Algebra

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Question:

I think of a number. Then I multiply by 2. Then I add 1. Then I divide the answer by 3. Then I add 2. My final answer is equal to the number I thought of at the beginning. Write an algebraic equation that represents this problem. Solve the equation to find out the number I started with.

Chinenye O.

Answer:

Note: Algebra is basically about solving for unknowns. In this case, the unknown is the number I started with. Follow the steps below to arrive at the algebraic equation: 1. Let the unknown number be x. 2. Multiply x by 2: 2x 3. Add 1: 2x + 1 4. Divide by 3: (2x + 1)/3 5. Add 2: (2x + 1)/3 + 2 6. Equate the expression to x: (2x + 1)/3 + 2 = x The algebraic equation that represents the problem is (2x + 1)/3 + 2 = x. Solving the equation: This equation can be solved using different methods. Let's explore three methods. Method 1: Steps to solve the equation; starting from (2x + 1)/3 + 2 = x 1. Subtract 2 from both sides of the equation: (2x + 1)/3 = x - 2 2. Multiply both sides of the equation by 3: 2x + 1 = 3(x - 2) 3. Open the parenthesis/bracket on the RHS of the equation: 2x + 1 = 3x - 6 4. Group like-terms together: 2x - 3x = -6 -1 5. Simplify: -x = -7 6. Divide both sides of the equation by -1: x = 7. Method 2: Steps to solve the equation; starting from (2x + 1)/3 + 2 = x 1. Multiply each expression in the equation by 3: (2x + 1) + 6 = 3x 2. Simplify: 2x + 7 = 3x 3. Group like-terms together: 2x - 3x = -7 4. Simplify: -x = -7 5. Divide both sides of the equation by -1: x = 7. Method 3: (Use this method if you're fraction savvy) Steps to solve the equation; starting from (2x + 1)/3 + 2 = x 1. Simplify the LHS of the equation: (2x + 1 + 6)/3 = x 2. Multiply both sides of the equation by 3: 2x + 1 + 6 = 3x 3. Simplify: 2x + 7 = 3x 4. Group like-terms together: 2x - 3x = -7 5. Simplify: -x = -7 6. Divide both sides of the equation by -1: x = 7. Check your answer (this is a good mathematical practice): Since the answer is x = 7, plug in 7 everywhere you see x in the original equation. The idea is to ascertain that the LHS and RHS of the equation are equal. The original equation is (2x + 1)/3 + 2 = x Let's evaluate the LHS: [2(7) + 1]/3 + 2 = [14 + 1]/3 + 2 = 15/3 + 2 = 5 + 2 = 7 Let's evaluate the RHS: x = 7 Since the LHS and RHS of the equation are both equal to 7, the answer is correct! Note: LHS means Left Hand Side. RHS means Right Hand Side. The decision to add, subtract, multiply or divide by a particular number at different steps since the solution is determined by the inverse operations.

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