What is the probability that a sum of 5 comes before a sum of 7 when rolling a pair of fair dice?
To solve the problem, we need to first determine the probability of rolling a sum of 5 and a sum of 7 with a pair of fair dice. The possible combinations for each sum are as follows: For a sum of 5: (1,4), (2,3), (3,2), (4,1) — a total of 4 combinations. For a sum […]
What is the length of an arc of 140 degrees when the circumference of a circle is 60 cm?
To find the length of an arc, we can use the formula: Arc Length = (θ/360) × C where θ is the angle in degrees and C is the circumference of the circle. Given: C (Circumference) = 60 cm θ (Angle) = 140 degrees Now we can substitute the values into the formula: Arc Length […]
Find the First Six Terms of the Sequence a1 = 7, an = an-1 + 6
To find the first six terms of the sequence defined by a1 = 7 and an = an-1 + 6, we can start with the first term and use the recursive formula to calculate the subsequent terms. First term (a1): Given as 7. Second term (a2): Using the formula, a2 = a1 + 6 = […]
Which function in vertex form is equivalent to f(x) = x² + 16x + 8?
To express the function in vertex form, we can start by rewriting the quadratic function. The given function is: f(x) = x² + 16x + 8 First, we need to complete the square. We take the coefficient of the x term, which is 16, divide it by 2 (getting 8), and then square it (resulting […]
What is the approximate value of x in the equation below log5(15) = x
To solve the equation log5(15) = x, we want to express the logarithm in a form that allows us to find an approximate value for x. Using the change of base formula, we can express the logarithm as follows: x = log(15) / log(5) Next, we can calculate the approximate values of the logarithms using […]
A line contains points M(1, 3) and N(5, 0). What is the slope of MN?
To find the slope of the line that passes through the points M(1, 3) and N(5, 0), we can use the formula for the slope (m) given two points (x1, y1) and (x2, y2): m = (y2 – y1) / (x2 – x1) Here, we can take M(1, 3) as (x1, y1) and N(5, 0) […]
How many solutions does this linear system have: y = x + 2 and 6x – 4y = 10?
To determine how many solutions this linear system has, we can solve the equations. The first equation is: y = x + 2 Now, we can substitute this expression for y into the second equation: 6x – 4(x + 2) = 10 Expanding this gives: 6x – 4x – 8 = 10 Combining like terms: […]
How to Evaluate the Integral of e^{ ext{cos} t} ext{sin}(2t) dt Using Substitution and Integration by Parts?
To evaluate the integral ∫ e^{ ext{cos} t} ext{sin}(2t) dt, we can start by using a substitution and then apply integration by parts. Step 1: Substitution Let’s use the substitution: u = ext{cos} t, which implies that du = – ext{sin} t \, dt or dt = - rac{du}{ ext{sin} t}. In terms of u, sin […]
Write an equation of the line that passes through the given two points
To find the equation of the line that passes through two points, we first need the coordinates of the points. Let’s call these points (x₁, y₁) and (x₂, y₂). The first step is to calculate the slope (m) of the line using the formula: m = (y₂ – y₁) / (x₂ – x₁) Once we […]
Simplify (x^(2/9))^(3/8)
To simplify the expression (x^(2/9))^(3/8), we will use the power of a power property of exponents, which states that when you raise a power to another power, you multiply the exponents. So, we will multiply the exponents: Base: x Exponent 1: 2/9 Exponent 2: 3/8 Now, we calculate: x^(2/9 * 3/8) To multiply the fractions: […]