Question

The total revenue for Jane's Vacation Rentals is given as the function R(x)=100x−0.25x^2, where x is the number of villas rented. What number of villas rented produces the maximum revenue?

Answer 1

@vocaloid

Answer 2

graphing would be the easiest way to solve this, simply graph and look where the vertex is

alternate method: when you have a quadratic equation

ax^2 + bx + c
a is the coefficient of the x^2 term, b is the coefficient of the x term, c is the constant

the x-coordinate of the vertex is x = -b/(2a)

Answer 3

if you're using mathway: enter the function 100x−0.25x^2, find the maximum/minimum value, take the x-coordinate since that's the # of villas

Answer 4

(200,10000)

Answer 5

good, and since it's asking for the # of villas, you would just take the x-coordinate 200

Answer 6

The total cost of producing a type of truck is given by C(x)=17000−90x+0.2x^2, where x is the number of trucks produced. How many trucks should be produced to incur minimum cost?

Answer 7

similar logic, enter the function, look for the minimum, take the x-value

Answer 8

(225,−8425)

Answer 9

good, and since x = number of trucks, you only want the x-coordinate 225

Answer 10

Consider the following function.

t(x)=(x−4)^2−9

Find the x-intercepts, if any. Express the intercept(s) as ordered pair(s).

How many x-intercepts does this function have ?

A. None
B. One
C. Two

Answer 11

it looks like 2

Answer 12

good

Answer 13

Consider the following function.

t(x)=(x−4)^2−9

Find two points on the graph of the parabola other than the vertex and x-intercepts.

A : ( , )

B : ( , )

Answer 14

it's in vertex form (x-h)^2 - k, where (h,k) is the vertex, so the vertex is (4,9)
to find the x-intercepts, we set t(x) = 0 and solve for x

0 = (x-4)^2 - 9
9 = (x-4)^2
square root of both sides

x-4 = 3
x - 4 = -3
so x = 7 or x = 1

****however*** it is asking for points **other than** the vertex and x-intercepts, so you have to plug in any x-value ***Except*** 1, 4, and 7

so you could plug in x = 2 and x = 3 to get your two points

Answer 15

x = 2 and x = 9 ?

Answer 16

yes, you could use x = 2 and x = 9, but you have to plug them into the function t(x)=(x−4)^2−9 to get the two points

Answer 17

calculate the value of the function at x = 2 and x = 9

Answer 18

t(2)=(9−4)^2−9

Answer 19

good, keep going

Answer 20

16

Answer 21

good, so (9,16) is one possible point

repeat with x = 2

Answer 22

okay, where i do i apply x = 2 in it ?

Answer 23

and is (9, 16) for A ?

Answer 24

just like you plugged in x = 9, plug in x = 2 into the function.

yes, you plug in point A as (9,16)

Answer 25

oh i see what you mean

Answer 26

23

Answer 27

check your arithmetic again

t(x)=(x−4)^2−9
x = 2, so

t(x) = (2-4)^2 - 9 = 4 - 9 = -5

so (2,-5) is the second point B

Answer 28

Consider the following function.

t(x)=(x−4)^2−9

Graph the parabola.

Answer 29

going off the points we've calculated so far:
the intercepts (1,0) and (7,0)
and the two points we just calculated: (9,16) and (2,-5), plot these points, draw the parabola through them

Answer 30

i'm not sure how this can be plotted on the graph on mine

Answer 31

Among all rectangles that have a perimeter of 84, find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.