What is the total weight of water in a full hemispherical tank with a diameter of 20 feet?
To find the total weight of the water in a full hemispherical tank, we first need to calculate the volume of the tank and then use the weight density of water to find the total weight. The formula for the volume V of a hemisphere is given by: V = (2/3) * π * r³ […]
How to Convert Standard Form to Vertex Form by Completing the Square
To convert a quadratic equation from standard form, which is given as y = ax² + bx + c, to vertex form, expressed as y = a(x – h)² + k, we can use the technique known as completing the square. Here’s a step-by-step approach: Start with the standard form: Let’s consider an example: y […]
What is the area of the region between the graph of y = 3x² + 2x and the x-axis from x = 1 to x = 3?
To find the area of the region between the graph of the function y = 3x² + 2x and the x-axis from x = 1 to x = 3, we will first calculate the definite integral of the function over this interval. The area A can be calculated using: A = ∫[1 to 3] (3x² […]
In triangle TRS, TZ is 3x inches and WZ is 2x + 3 inches. What is WZ in inches?
To find WZ in inches, we need to first express both lengths in terms of x and then solve the equation to find the value of WZ. We know that TZ is given as 3x inches and WZ is given as 2x + 3 inches. However, we need to relate these lengths to find a […]
With an interest rate of 6 percent, what is the present value of $100 next year?
To find the present value of $100 that you will receive next year at an interest rate of 6 percent, we can use the present value formula: Present Value (PV) = Future Value (FV) / (1 + r)^n Where: PV = Present Value FV = Future Value ($100 in this case) r = interest rate […]
What is the 22nd term of the arithmetic sequence where a1 is 8 and a9 is 56?
To find the 22nd term of the arithmetic sequence, we first need to identify the first term and the common difference. We know that the first term (a1) is 8, and the ninth term (a9) is 56. The formula for the n-th term of an arithmetic sequence is: a_n = a1 + (n – 1) […]
Find the Point on the Line y = 4x + 3 That Is Closest to the Origin
To find the point on the line y = 4x + 3 that is closest to the origin (0, 0), we can use the concept of minimizing the distance between the point on the line and the origin. The distance D from the origin to a point (x, y) on the line can be given […]
What is the average value of the function f(x) on the given interval [0, 16]?
To find the average value of the function f(x) over the interval [0, 16], we first need to clarify the function. Let’s assume the function is given as f(x) = x. The formula for the average value of a continuous function f over an interval [a, b] is given by: Average Value = (1/(b – […]
How do you find the points on the surface y² = 64xz that are closest to the origin?
To find the points on the surface defined by the equation y² = 64xz that are closest to the origin, we can use the method of Lagrange multipliers or minimize the squared distance from the origin. The squared distance from a point (x, y, z) to the origin is given by D = x² + […]
What is the other factor of the polynomial 2x² + 5x – 3 if one factor is x – 1?
To find the other factor of the polynomial 2x² + 5x – 3 given that one of the factors is x – 1, we can use polynomial long division or synthetic division. First, we can rewrite 2x² + 5x – 3 as: (x – 1)(Ax + B) We need to find the constants A and […]