Answer:
Step-by-step explanation:
<u>Optimizing With Derivatives
</u>
The procedure to optimize a function (find its maximum or minimum) consists in
:
- Produce a function which depends on only one variable
- Compute the first derivative and set it equal to 0
- Find the values for the variable, called critical points
- Compute the second derivative
- Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum
We know a cylinder has a volume of 4 
. The volume of a cylinder is given by
Equating it to 4
Let's solve for h
A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is
Replacing the formula of h
Simplifying
We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero
Rearranging
Solving for r
![\displaystyle r=\sqrt[3]{\frac{4}{\pi }}\approx 1.084\ feet](https://tex.z-dn.net/?f=%5Cdisplaystyle%20r%3D%5Csqrt%5B3%5D%7B%5Cfrac%7B4%7D%7B%5Cpi%20%7D%7D%5Capprox%201.084%5C%20feet)
Computing h
We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative
We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.
The minimum area is
Solution:
Total number of Days that Pooja's plant survived= 2 + 98 = 100 Days=(t)
Longest Height obtained by plant = 30 cm= h
As the two , i.e number of Days lived by plant and it's longest height obtained are directly proportional.
We will solve it by unitary method.
100 Days = 30 cm
1 cm = Days
If height of the plant is h cm ,and amount of time taken is t days, then
h(t) cm = ,for t=0,1,2,3,4,5,6....we get different values of h.
To find the total amount Mrs. Johnson gave away, you need to know how much she gave the son. Because the ratio of the money given is 5:6, the amount of money he received has to be 5/6 of the money the daughter received.
So, the son received 5/6 of 2400.
To find this, multiply 5/6 x 2400 to get 2000.
The daughter received 2400 and the son received 2000, so the total amount Mrs. Johnson gave away is 4400 dollars.
Given:
The equations are
To find:
Which function is changing more quickly.
Solution:
The slope intercept form of a line is
Where m is slope and b is y-intercept.
On comparing with the slope intercept form, we get
On comparing with the slope intercept form, we get
Since, 
, it means the absolute rate of change of second function is greater than the first function.
Therefore, the second function is changing more quickly.
slope intercept form: y=mx+b
y=2x-3
slope is also m
b is the y-intercept
answer:
slope is 2
y-intercept is -3
