Pipe for a water line must be installed from a main water line at point A to a building on Hontoon Island State Park at point B as shown in the figure. The cost to install water pipe over land is $ 10 per foot and the cost to install pipe under water is $ 20 per foot a. Write an expression in terms of 6 to represent the total cost c (in dollars) to lay pipe from point A to point B . b. Use the TABLE function on a calculator to find the cost for 6 = 20 ° , 25 ° , 30 ° , 35 ° , and 40 ° . Round to 1 decimal place. c. Which angle from part (b) yields the least cost? d. Using calculus, we can show that the angle needed to minimize the total cost is a solution to the equation 4000 sec θ tan θ − 2000 sec 2 = 0 . Solve the equation for 6 , where 0 ° < θ < 90 ° .
Pipe for a water line must be installed from a main water line at point A to a building on Hontoon Island State Park at point B as shown in the figure. The cost to install water pipe over land is $ 10 per foot and the cost to install pipe under water is $ 20 per foot a. Write an expression in terms of 6 to represent the total cost c (in dollars) to lay pipe from point A to point B . b. Use the TABLE function on a calculator to find the cost for 6 = 20 ° , 25 ° , 30 ° , 35 ° , and 40 ° . Round to 1 decimal place. c. Which angle from part (b) yields the least cost? d. Using calculus, we can show that the angle needed to minimize the total cost is a solution to the equation 4000 sec θ tan θ − 2000 sec 2 = 0 . Solve the equation for 6 , where 0 ° < θ < 90 ° .
Solution Summary: The author calculates theta to represent the total cost to lay the pipe from a main water line in HintonIsland State Park.
Pipe for a water line must be installed from a main water line at point A to a building on Hontoon Island State Park at point B as shown in the figure. The cost to install water pipe over land is $
10
per foot and the cost to install pipe under water is $
20
per foot
a. Write an expression in terms of
6
to represent the total cost c (in dollars) to lay pipe from point A to point B.
b. Use the TABLE function on a calculator to find the cost for
6
=
20
°
,
25
°
,
30
°
,
35
°
,
and
40
°
. Round to 1 decimal place.
c. Which angle from part (b) yields the least cost?
d. Using calculus, we can show that the angle needed to minimize the total cost is a solution to the equation
4000
sec
θ
tan
θ
−
2000
sec
2
=
0
. Solve the equation for
6
, where
0
°
<
θ
<
90
°
.
1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.
Elementary Statistics: Picturing the World (7th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Problems on Area and Circumference of Circle| Basics of Circle| Questions on Circle||BrainPanthers; Author: Brain Panthers;https://www.youtube.com/watch?v=RcNEL9OzcC0;License: Standard YouTube License, CC-BY