Find a Cartesian Equation for the Curve r = 8 sin(8θ) cos(θ)
To convert the polar equation r = 8 sin(8θ) cos(θ) into a Cartesian equation, we can use the relationships between polar and Cartesian coordinates: x = r cos(θ) and y = r sin(θ). First, let’s rewrite the polar equation: r = 8 sin(8θ) cos(θ) Now, we know that: sin(8θ) = rac{y}{r}, cos(θ) = rac{x}{r} Substituting […]
A number is divisible by both 5 and 12. By which other number will that number be always divisible?
If a number is divisible by both 5 and 12, it is also divisible by their least common multiple (LCM). To find the LCM, we first consider the prime factorization of both numbers. The number 5 is a prime number, so its factorization is simply 5 itself. The number 12 can be factored into primes […]
What is the area of a sector with a central angle of 210 degrees and a diameter of 46m?
To find the area of a sector, we can use the formula: Area = (θ / 360) × π × r² Where: θ is the central angle in degrees r is the radius of the circle First, we need to determine the radius from the diameter. The diameter of the circle is 46 meters, so […]
Find the cosine and sine of 180 degrees. Round your answers to the nearest hundredth if necessary.
To find the cosine and sine of 180 degrees, we can use the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane. At 180 degrees, the point on the unit circle is (-1, 0). This means: The cosine of 180 degrees is the x-coordinate of […]
What is the missing term that makes these ratios equivalent 153: ? 315?
To find the missing term that makes the ratios equivalent, we need to set up an equation based on the property of equivalent ratios. The ratio 153 to x should be equal to the ratio x to 315. We can write this relationship as: 153/x = x/315 Next, we can cross-multiply to solve for x: […]
How many integers from 0 to 50 inclusive have a remainder of 1 when divided by 3?
To find the integers from 0 to 50 that have a remainder of 1 when divided by 3, we first identify the numbers that satisfy this condition. An integer, n, gives a remainder of 1 when divided by 3 if it can be expressed in the form: n = 3k + 1 where k is […]
If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f(4) and explain.
To solve this problem, we’re dealing with a tangent line to the function f(x) at the point (4, 3). We need to find the function value f(4) and consider the slope of the tangent line. First, since the point (4, 3) lies on the function f, we know that f(4) = 3. Next, we need […]
What is the rough estimate and closer estimate of 4125, 139, 62?
To find a rough estimate and a closer estimate of the numbers 4125, 139, and 62, we can round each number to make calculations simpler. Rough Estimate: For a rough estimate, we can round each number to the nearest hundred or thousand, depending on their size. 4125 rounds to 4100 139 rounds to 100 62 […]
If the surface area of a cell increases by a factor of 100, what will happen to the volume of that cell?
When the surface area of a cell increases by a factor of 100, the volume of the cell will increase by a factor of 1000. To understand this, we need to consider the relationship between surface area and volume. The surface area (SA) of a cell is proportional to the square of its dimensions, while […]
What constant term should be used to complete the square x² + 5x – 7?
To complete the square for the quadratic expression x² + 5x – 7, we need to focus on the x² and 5x part. The general process involves finding a constant that makes the expression a perfect square trinomial. First, we look at the coefficient of the x-term, which is 5. To find the constant term, […]