Real Life Applications of Calculus You Didnt Know About

application number one assume you own company that manufactures boxes and you are also an engineer you are provided square cardboard sheet like this whose sides are of length 6 ft we need to make box using this cardboard sheet whose top is open like shown here and how can you construct this box first cut out square from each of the four corners of the sheet like this and then one by one Bend up the sides of this sheet our main goal here is to find the value of the length of this cut off piece which maximizes the volume of the box so can you solve it after looking at this question you are confused and you go to your best friend calculus for help he says don't worry in order to achieve the maximum volume of this open box let's define the process mathematically the original Square sheet has side length of 6 ft let be the side length of the small square that we cut from each corner so what will be the length of this piece it will simply be 6 - - or 6 - 2x right so all four of them will be of this length and thus we have the base of this open box equal to square whose side length is 6 - 2x so what will be its area the area of the square base will be equal to 6 - 2x² isn't it now what will be the volume of this box see the height of this box will be the same as the size of this piece which was cut off from this Square so the height will be equal to therefore the volume of the box or of will be equal to the area of the base time its height or 6 - 2 x^ 2 * to find the value of that gives the maximum volume we take the derivative of of with respect to and set it to zero and then solve for if you are confused why we do it like this then have already made video on the same and the link is in the description but complete this video first now using the product rule for derivatives we get the derivative of of as this again if you don't know what product rule is have already made video on the same and the link is in the description again complete this video first now will not bore you by solving quadratic equation the roots of this equation are = 3 and = 1 now obviously if = 3 ft then 6 - 2x will become 0 and this is not practical so we have = 1T as the final answer thanks to calculus we've optimized the Box designed to achieve the maximum possible volume with precise mathematical reasoning application number two now once the product is made the next step is to Market your product isn't it imagine you are selling the box you made right now you need to decide how much to charge for each item if you set the price too high no one will buy it if you set the price too low you might sell lot but not make enough profit marketing research gives us some useful information if we set the price at $10 nobody will buy the boxes however for each $1 we lower the price we can sell 500 more boxes this means if we set the price at $9 we can sell 500 boxes if we set the price at $8 we can sell1 th000 boxes and so on every business has costs some costs stay the same no matter how many items we sell for example buying the machine these are called fixed costs in this case the fixed cost is $33,000 other costs change depending on how many items we make these are called variable costs in this case each box costs $2 to produce so what what is the best price to charge in order to make the maximum profit can you solve it let's call the price of the box as then the number of boxes we can sell depends on the formula for the number of boxes sold or of will then be 10 minus then multiplied by 500 so if is 9 then this will be 10 - 9 * 500 or 500 boxes sold and if is 8 then this will be 10 - 8 * 500 or 1,000 boxes sold right now the total money we make is called Revenue our revenue is just the price of one box which is multiplied by the number of boxes sold which is of so the revenue is * 10 minus then multiplied by 500 expand it to get 5,000 minus 500 p² noise now let's talk about costs as mentioned before every business has costs for our case the total cost is fixed cost or $3,000 plus the cost of making each item which is two multiplied by the number of items sold or of this means the total cost is 3,000 + 2 * 10 - * 500 expand this to get the cost as 13 ,000 - 1000p great now we can finally talk about profit profit is the money left after we subtract the costs that our business incurs from the revenue so profit is revenue minus total cost or this minus this which after adding the like terms gives us this now our goal is to find the price that makes profit as large as possible you know what to do to find the best price we take the derivative of profit and set it to zero derivative of this will be 6,000 - th000 * and this equals Zer so we get equal 6,000,000 or $6 this means the best price to charge for each box will be $6 awesome application number three first you optimized the boxes then you marketed it figuring out the best price to Max maximize profits your business is booming now it's time for the next challenge getting the product to customers efficiently one day special order comes in an elegant handcrafted table inside this cardboard box for penthouse apartment your delivery team arrives but there's problem the apartment building has high fence which is 8 ft tall and the main entrance is blocked due to ongoing construction the the only way in is to lift the table over the fence and pass it through balcony that's 3 ft behind it as you can see we can use ladder like this but we can also keep the ladder like this and this and many other possible ways but we need ladder that is just long enough to reach over the fence and touch the balcony without being longer than necessary in order to avoid the extra cost so what is the shortest possible ladder that can get the job done can you solve it okay let be the length of the ladder which we need to minimize now let the ladder touch the ground at some point ft away from the wall so this will be - 3 also let the ladder reach the balcony at height ft above the ground first we will find relationship between and if we let this angle be Theta then what will be the tangent of theta in this case so for this triangle it will be 8 /x - 3 right and for this triangle it will be overx so equating both of them together like this will give us = 8x /x - 3 amazing now our job is to minimize right next we will use Pythagoras Theorem to find the relation between and we get = x² + h² substitute from here to get this so equal the square root of this we will simply differentiate with respect to and then set it to zero see the main objective of this video is not to learn how to differentiate but instead look at the real life application of calculus so we will use an online calculator to find its derivative which have already done for you it will be this therefore or setting it to zero gives us this numerator equals 0 or this equals 0 this gives us = 0 and this equals 0 look here we have this side length as - 3 so cannot be equal to and it must be greater than 3 now this equation gives 1 equals this or - 3 Cub = 192 take cube root on both sides to get - 3 = cubot of 192 so = 3 + cube root of 192 or nearly 8.77 substitute it here to get as approximately 15 ft and that's it the shortest possible ladder that could reach over the 8T fence to the wall 3 ft behind was about 15 ft long isn't that amazing finally let's move on to application number four after making good amount of money from your business you decide to take break and spend time with your family you all decide to go watch movie at luxurious theater you grab your popcorn find perfect seat and then wait is this really the best seat you start thinking where should you sit to get the best possible view of the screen the movie screen is 20 ft high and is positioned 10 ft above the floor you are sitting ft away from the screen your goal is to maximize the viewing angle Theta between the top and bottom of the screen so can you solve it the total viewing angle Theta is the angle between your line of sight to the bottom of the screen and your line of site to the top of the screen let us introduce another angle Alpha like this it follows from basic trigonometry that tan of alpha = 10 /x so Alpha equal tan inverse of 10 /x in the similar fashion we get tan of alpha + Theta = 20 + 10 or 30 /x so Alpha + Theta = tan inverse of 30 /x so theta equals this minus Alpha substitute Alpha here to get Theta equal Tan in inverse of 30 /x minus tan inverse of 10 /x again if we want to maximize this Alpha you know what to do right just differentiate Theta with respect to and set it to zero the derivative of theta with respect to will be this and when we set it to zero we get this equals this now cross multiply to get this take 10 x² here to get 20 x² = 6,000 or x² = 300 which gives us = 10 * < TK 3 great now substitute it here to get Theta equal 30° so to get the best possible Movie experience you should sit about 10 < tk3 ft or 17.3 ft from the screen which will give you the maximum view angle of 30° thanks to calculus you can now enjoy your movie in the most immersive way remember calculus is everywhere now these types of videos take time to edit which drains all the energy and therefore in order to recharge myself need 10,000 likes on this video so that can come up with more awesome content for you all so good