Tutor profile: Gaurav M.
If a square of side length 8 is perfectly inscribed within a circle such that the four corners of the square lie on the circle's perimeter, what is the area of the circle?
To begin, we draw a picture of the problem and analyze the situation. We are given the side length of the square to be 8. However, we need the area of the circle. If a line is drawn from one corner of the square to the opposite corner, this line represents the diameter of the circle, which can be used to find the radius, and subsequently, the area of the circle. When this line is drawn, it divides the square into two right triangles with a hypotenuse of the diameter of the circle. Using the Pythagorean theorem, if 8^2 + 8^2 = x^2, x is equal to sqrt(128) or 8sqrt(2). This value represents the diameter of the circle. To obtain the radius, we divide this value by 2 to get 4sqrt(2). Thus, the area of the circle is pi*(4sqrt(2))^2 = 32 pi.
Solve the following equation for x: 6x + 4 = (x/2) - 7.
To begin, work to isolate x, or transferring the variable terms on one side and the numerical constants to the other side. That gives us 6x - (x/2) = -7 - 4, which simplifies to 6x - (x/2) = -11. To add the terms on the left side of the equation, we can multiply both sides of the equation by 2 to get rid of the denominator in x/2: 2*(6x - (x/2)) = 2*(-11), simplifying to 12x - x = -22 and then 11x = -22. Dividing both sides by 11 gives us x = -2. This answer can be checked by substituting this value into the original equation: 6(-2) + 4 = (-2/2) - 7. Both sides simplify to -8, so we are sure x = -2 is the correct answer.
Solve for x in the equation x^2 + 6x = -1.
This equation can be solved using the "complete the square" method. Adding 1 to both sides gives us x^2 + 6x + 1 = 0. We start by taking the coefficient of x, which is 6, dividing by 2, and squaring it to get 9 (6/2 = 3, 3^2 = 9). To complete the square on the left side of the equation by obtaining x^2 + 6x + 9, add 8 to both sides to get x^2 + 6x + 1 + 8 = x^2 + 6x + 9 = 8. By algebraic principles, x^2 + 6x + 9 = (x+3)^2. Thus, (x+3)^2 = 8. Taking the square root of both sides gives us x+3 = (+/-) sqrt(8). Thus, the solutions for x are -3 + sqrt(8) and -3 - sqrt(8).
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