Math Problems: Profit, Division, And Trigonometry

Hey guys! Let's dive into some math problems today. We'll tackle questions about profit calculations, dividing money, and even a bit of trigonometry. Don't worry, it's not as scary as it sounds! I'll break everything down step-by-step so you can follow along easily. We will find out net profit and solving the trigonometry equation to get the correct answer. So grab your calculators (or your brains!) and let's get started!

(a) Calculating Net Profit from Project Earnings

Alright, so imagine a contractor who's just wrapped up a project and earned a cool Ksh. 500,000. Now, like any good businessperson, they want to know how much profit they actually made. This is where we come in to help them figure out the net profit. They've given us some key information: the labor costs ate up 40% of their earnings, and the materials cost another 35%. So, how do we find the profit? It's all about understanding what goes into the costs and then subtracting those costs from the total earnings. This will help you get a great understanding of your net profit. Let's break it down into easy steps, okay?

First, let's figure out the total cost. Labor is 40%, and materials are 35%. That means the combined cost of labor and materials is 40% + 35% = 75%. This is the total percentage of their earnings that went towards these expenses. Now, to find out the actual amount spent on labor and materials, we multiply their total earnings by 75%. So, 500,000 * 0.75 = 375,000. This means the contractor spent Ksh. 375,000 on labor and materials.

Next comes the fun part: calculating the net profit. The net profit is simply the total earnings minus the total costs. In this case, that's Ksh. 500,000 (total earnings) - Ksh. 375,000 (total costs) = Ksh. 125,000. So, the contractor's net profit from the project is a sweet Ksh. 125,000! That's the money they get to keep after covering all their expenses. Pretty neat, huh? Understanding how to calculate net profit is super important for anyone in business, as it helps them see how well their business is doing. Always remember to break down the cost for you to understand better.

To make sure you understand, let's recap: We found the total cost (75% of earnings), calculated the actual cost amount, and then subtracted that cost from the total earnings to get the net profit. Simple as that! By following this method, anyone can easily calculate their net profit.

Additional considerations for calculating net profit

Besides labor and materials, other expenses can affect the net profit. These might include things like transportation, equipment rental, or even the contractor's own salary. If we wanted to get really detailed, we'd need to factor in all of these expenses. We'd add them to the labor and material costs to find the total costs, and then subtract that total from the earnings. Another factor to consider is taxes. In reality, the contractor would also have to pay taxes on their profit, which would further reduce the amount they take home. So, the actual net profit might be slightly lower than what we calculated, but the principle remains the same. The basic idea is to subtract all the costs from the total earnings.

But for this problem, we kept it simple, focusing on labor and materials. Now, let's move on to the next part, where we learn how to divide the profit to be fair. It's really good to see how you split the money, whether it's for profit sharing in a business or maybe splitting the bill with friends.

(b) Dividing Money in a Given Ratio

Alright, now let's change gears and talk about dividing money. Imagine we have Ksh. 72,000 to divide among three people: A, B, and C. They've agreed to split the money in a specific ratio: 5:3:4. This means for every Ksh. 5 A gets, B gets Ksh. 3, and C gets Ksh. 4. We want to find out how much money each person receives. So, how do we do it? It's all about understanding the ratio and distributing the amounts fairly, based on the net profit of the overall investment. This is a very common problem that involves dividing something (like money, resources, or even time) in proportion to certain numbers. Let's break it down so it's easy to grasp. Dividing money in a ratio is all about determining how each person or entity should get.

First, we need to find the total parts of the ratio. In this case, the ratio is 5:3:4. To find the total parts, we simply add the numbers together: 5 + 3 + 4 = 12. This means that the total amount of money (Ksh. 72,000) is being divided into 12 parts. Next, we determine the value of one part. To do this, we divide the total amount of money by the total parts: Ksh. 72,000 / 12 = Ksh. 6,000. This means that each 'part' of the ratio is worth Ksh. 6,000. Now that we know the value of one part, we can figure out how much each person gets. A gets 5 parts, so A receives 5 * Ksh. 6,000 = Ksh. 30,000. B gets 3 parts, so B receives 3 * Ksh. 6,000 = Ksh. 18,000. C gets 4 parts, so C receives 4 * Ksh. 6,000 = Ksh. 24,000.

So, the final distribution is: A gets Ksh. 30,000, B gets Ksh. 18,000, and C gets Ksh. 24,000. If you want to double-check your work, add up the amounts each person receives. It should equal the total amount of money, Ksh. 72,000. And yes, 30,000 + 18,000 + 24,000 = 72,000! Great job, guys! This method can be used to divide anything, whether it's splitting the profits from a company or resources across different departments. Understanding ratios can be super helpful in a lot of situations.

The importance of understanding ratios

Ratios are essential in many areas of life, not just mathematics. They help us compare different quantities and understand their relationships. In cooking, ratios are crucial for scaling recipes. In business, they're used to analyze financial statements and make decisions. Even in everyday life, we use ratios to compare prices, distances, and so much more. This understanding of ratios is important in order to understand how to divide the total net profit of the investment. Ratios provide us a simple way of distributing and making it easy to understand.

For example, if a business has several partners and decides to split profits according to their initial investments, ratios help to fairly calculate each person's share. If one partner invested more capital, they would get a larger portion of the profits. This would be reflected in the ratio. Ratios will ensure that the distribution is transparent and equitable, which is crucial for maintaining trust and partnership. So, as you can see, understanding ratios is a valuable skill that can be applied in numerous real-life scenarios. It is very useful, so take your time and understand this concept.

(a) Solving Trigonometric Equations

Now, let's switch gears and delve into a bit of trigonometry! We need to solve for x in the equation 5sin(x) = 2.5. This might look a bit intimidating if you haven't seen trigonometry in a while, but trust me, we'll break it down step by step to get the answer. We'll use the principles of trigonometry to find the value(s) of x that make the equation true. Solving trigonometric equations involves manipulating trigonometric functions, identities, and inverse functions. To start, let's isolate the sin(x) term.

First, we need to isolate the sin(x) term. The equation is 5sin(x) = 2.5. To get sin(x) by itself, we divide both sides of the equation by 5. This gives us sin(x) = 2.5 / 5, which simplifies to sin(x) = 0.5. Now, we need to find the angle(s) whose sine is 0.5. This is where we need to remember our unit circle or use a calculator with an inverse sine (sin⁻¹) function. The inverse sine function (also known as arcsin) gives us the angle whose sine is a specific value. So, we'll use sin⁻¹(0.5). Using a calculator, sin⁻¹(0.5) = 30 degrees (or π/6 radians). This means one solution for x is 30 degrees. However, there can be multiple solutions in trigonometry because the sine function is periodic. Meaning, it repeats its values over and over.

To find other possible solutions, we need to understand the properties of the sine function. The sine function is positive in the first and second quadrants of the unit circle. The first quadrant gives us 30 degrees. To find the angle in the second quadrant, we subtract 30 degrees from 180 degrees (since the second quadrant lies between 90 and 180 degrees). So, 180 degrees - 30 degrees = 150 degrees. Therefore, another solution for x is 150 degrees. If we were working in radians, the solutions would be π/6 and 5π/6. So, the solutions to the equation 5sin(x) = 2.5 are x = 30 degrees and x = 150 degrees (or π/6 and 5π/6 radians). Awesome! We’ve successfully solved a trigonometric equation.

More about trigonometric equations

This was a relatively simple trigonometric equation. More complex equations might involve multiple trigonometric functions, identities, and algebraic manipulations. For example, you might need to use the Pythagorean identity (sin²(x) + cos²(x) = 1) or other trigonometric relationships to simplify the equation. The key to solving more complex trigonometric equations is to know your trigonometric identities well and to be comfortable with algebraic manipulations. Make sure to double-check your answer when working with trigonometric equations, as there can sometimes be multiple solutions, and it is easy to miss one. Always be careful to determine the range of values for which you're solving the equation, as this can affect the number of possible solutions. Remember that the sine function, as well as the cosine function, and tangent function, is periodic, which means it repeats its values in a regular pattern.

(b) From a Point on

I am sorry, but the question appears to be incomplete. Please provide the complete question. Without the complete question, I am unable to solve it. Please ensure to provide all of the details for the problems to make sure that they are solvable. If you can provide the question, I will gladly work on it!

I hope this was helpful! Feel free to ask if you have any further questions, and have a good day!