![austinmontgomep7foxp](/avatars/32172.jpg)
austinmontgomep7foxp
05.05.2020 •
Mathematics
What is m<1 in this parallelogram?
Solved
Show answers
More tips
- G Goods and services How to Choose a Video Camera: Tips from Professionals...
- F Family and Home How to Remove Tar Stains: Tips and Recommendations from Experts...
- H Health and Medicine Novomin: What is it and how to use it?...
- P Philosophy Unbelievable stories of encounters with otherworldly forces...
- L Leisure and Entertainment How to Choose the Perfect Gift for Men on February 23rd?...
- H Health and Medicine How to Treat Whooping Cough in Children?...
- H Health and Medicine Simple Ways to Lower Cholesterol in the Blood: Tips and Tricks...
- O Other How to Choose the Best Answer to Your Question on The Grand Question ?...
- L Leisure and Entertainment History of International Women s Day: When Did the Celebration of March 8th Begin?...
- S Style and Beauty Intimate Haircut: The Reasons, Popularity, and Risks...
Answers on questions: Mathematics
- M Mathematics Explain how 2/3 and 7/8 are not equivalent...
- S Spanish I need this asap (also use level 2 spanish cus im in highschool)...
- M Mathematics If we dilate a triangle with a scale factor of 3 how can we calculate the side lengths of the new triangles...
- H History Following world war ii, what name was given to nations that played a dominant economic, poltical, and military role in the world?...
- M Mathematics a given population is known to be 800 in the year 2000 for particular bird in a region. The expected rate of growth to the population is 2% every two years. Determine...
Ответ:
They all start the same way: Let's call the length of the rectangle b and the width a. The perimeter of the rectangle (the distance all the way around -- that is, the length of the "fence") is 2a + 2b. Since the barn is used as one of the sides (let's say b) we can subtract b ... we don't need fencing for this side. That makes the perimeter 2a + b. Since we have 300 feet of fencing we set these equal: 2a + b = 300.
We can solve for b above by subtracting 2a from both sides of the equation so we get: b = 300 - 2a.
We are asked to maximize the area. The area of a rectangle is length times width. In this case, (a)(b) and since we just found an expression for b, we can substitute this to obtain: Area =
METHOD 1: Using Calculus
We are looking for the highest point of the equation
First we find the derivative of the function.
First we find the derivative of the function.
This means that the maximum value (the largest area) occurs when a = 75.
At a = 75 the area is given by
What is the largest area? Let's us substitute 75 for a in the equation of the area. This gives:
METHOD 2: Graphing
We know that the area is given by. Area =
This is a quadratic equation (the highest exponent of x is 2) and so the equation of a parabola (like the letter U or an upside down U). Since the leading coefficient is negative (-2) we know it opens downward (called concave down -- think of a frown). We are looking for the x-coordinate of the highest point called the vertex. We can find this by remembering that the x-coordinate of the vertex is given by
The y-coordinate of the vertex is found by substituting 75 for x in the equation for the area.
Since we let x be the width of the rectangle. We know that the width is 75. The area of the rectangle we called y. So the maximum area is 11250 square feet which occurs when the width of the rectangle is 75.
You could have gotten to this point by generating a list of points for the parabola and graphing it carefully on graph paper (though it might not have been exact depending on how you scale your axis).
METHOD 3: Guess and Check
The area of the rectangle is given by Area =