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622ghernandezc
14.04.2021 •
Mathematics
Please help me!
Organize your data and find the rate of change.
Create a table of the measurements for your data. Label the forearm measurements as your input and the foot measurements as your output.
Select two sets of points and find the rate of change for your data.
Describe your results. If you had to express this relation as a verbal statement, how would you describe it?
Here are the measurements
__Mom__Dad__Brother___Me___Friend__
Forearms| 11in | 1FT | 8in | 10in | 11in | INPUT
Foot length | 8in | 9in | 6in | 9in | 9in | OUTPUT
Solved
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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 =