Answer:
[tex]\displaystyle f(x)=\frac{3}{2}x^3-\frac{27}{2}x^2+36x-21[/tex]
Where:
[tex]\displaystyle a=\frac{3}{2}, \, b=-\frac{27}{2}, \, c=36, \text{and } d=-21[/tex]
Step-by-step explanation:
We are given a cubic function:
[tex]f(x)=ax^3+bx^2+cx+d[/tex]
And we want to find a, b, c and d such that the function has a relative maximum at (2, 9); a relative mininum at (4, 3); and an inflection point at (3, 6).
Since the function has a relative maximum at (2, 9), this means that:
[tex]f(2)=9=a(2)^3+b(2)^2+c(2)+d[/tex]
Simplify:
[tex]8a+4b+2c+d=9[/tex]
Likewise, since it has a relative minimum at (4, 3):
[tex]f(4)=3=a(4)^3+b(4)^2+c(4)+d[/tex]
Simplify:
[tex]64a+16b+4c+d=3[/tex]
We can subtract the first equation from the second. So:
[tex](64a+16b+4c+d)-(8a+4b+2c+d)=(3)-(9)[/tex]
Simplify:
[tex]56a+12b+2c=-6[/tex]
Divide both sides by two. Hence:
[tex]28a+6b+c=-3[/tex]
Relative minima occurs only at the critical points of a function. That is, it occurs whenever the first derivative equals zero.
Find the first derivative. We can treat a, b, c and d as constant. Hence:
[tex]f'(x)=3ax^2+2bx+c[/tex]
Since it has a minima at (2, 9), it means that:
[tex]f'(2)=3a(2)^2+2b(2)+c=0[/tex]
Thus:
[tex]12a+4b+c=0[/tex]
(We will only need one of the two points to complete the problem.)
Inflection points occurs whenever the second derivative of a function equals zero. Find the second derivative:
[tex]f''(x)=6ax+2b[/tex]
Since there is a inflection point at (3, 6):
[tex]18a+2b=0\Rightarrow 9a+b=0[/tex]
Solve for b:
[tex]b=-9a[/tex]
Substitute this into the above equation:
[tex]12a+4(-9a)+c=0[/tex]
Solve for c:
[tex]c=24a[/tex]
Substitute b and c into the previously acquired equation:
[tex]28a+6(-9a)+(24a)=-3[/tex]
Solve for a:
[tex]\displaystyle -2a=-3\Rightarrow a=\frac{3}{2}[/tex]
Solve for b and c:
[tex]\displaystyle b=-9\left(\frac{3}{2}\right)=-\frac{27}{2}\text{ and } c=24\left(\frac{3}{2}\right)=36[/tex]
Using either the very first or second equation, solve for d:
[tex]\displaystyle 8\left(\frac{3}{2}\right)+4\left(-\frac{27}{2}\right)+2(36)+d=9[/tex]
Hence:
[tex]d=-21[/tex]
Hence, our function is:
[tex]\displaystyle f(x)=\frac{3}{2}x^3-\frac{27}{2}x^2+36x-21[/tex]
Steve thinks he can drive legally 30 minutes after he drinks 5 beers. The legal limit is BAC = 0.08. Give a 90% prediction interval for Steve’s BAC. Can he be confident he won’t be arrested if he drives and is stopped?
Answer: Hello your question has some missing data attached below is the missing data
How well does the number of beers a student drinks predict his or her blood alcohol content (BAC)? Sixteen student volunteers at Ohio State University drank a randomly assigned number of 12-ounce cans of beer. Thirty minutes later, a police officer measured their BAC. Here are the data. The students were equally divided between men and women and differed in weight and usual drinking habits. Because of this variation, many students don’t believe that number of drinks predicts BAC well.
answer:
prediction interval : (0.040 , 0.114)
Step-by-step explanation:
Given data:
Confidence level = 90%
Legal limit ( BAC ) = 0.08
solution
sample size = 16
Degree of freedom ( df ) = 14
critical t value = 1.761
X = 4.81
Σ(x-x)² (Sx) = 72.44
also standard error of estimates = 0.0204
Y= -0.01270 + 0.01796 * 5 = 0.077
given that ; the predicted value of Y at x = 5
Considering individual response Y
standard error = 0.0211
margin of error = 1.761 * 0.021 = 0.0371
Hence the limits of the prediction interval is :
Lower limit = 0.077 - 0.037 = 0.040.
Upper limit = 0.077 + 0.037 = 0.114
Finally
90% prediction interval = (0.040 , 0.114)
Is u=−12 a solution of 8u−1=6u?
Answer:
No, -12 is not a solution.
Step-by-step explanation:
8u-1=6u
8(-12)-1=6(-12)
-96-1=-72
-97=-72
Untrue, to it’s not a solution
Indigo Company had cash sales of $78,120 (including taxes) for the month of June. Sales are subject to 8.5% sales tax. how to record the sale ?
Answer:
Step-by-step explanation:
Sales revenue x (100%+sales tax rate) = total sales including tax
sales revenue x 1.085 (100% + 8.5%) = 78120
sales revenue = 78120/1.085
sales revenue = 72000
total sales including tax - sales revenue = sales tax
78120 = 72000 = 6120
Journal entry:
cash. 78120
Sales revenue. 72000
Sales tax payable. 6120
What is the area of the right triangle with sides 10,26 and 24
Answer:
[tex]\boxed {\boxed {\sf 120 \ units^2}}[/tex]
Step-by-step explanation:
We are asked to find the area of a triangle. The formula for calculating this is:
[tex]a= \frac{1}{2} bh[/tex]
This is a right triangle, so the base and height are the legs of the triangle. The 2 smallest sides are the legs because the longest side is the hypotenuse. Since the side lengths are 10, 26, and 24, the base and height must be 10 units and 24 units.
b= 10 unitsh= 24 unitsSubstitute these values into the formula.
[tex]a= \frac{ 1}{2} ( 10 \ units)(24 \ units)[/tex]
Multiply the numbers in parentheses.
[tex]a= \frac{1}{2}(240 \ units^2)[/tex]
Multiply by 1/2 or divide by 2.
[tex]a= 120 \ units^2[/tex]
The area of the triangle is 120 units squared.
Hãy tìm hàm gốc f(t) có hàm ảnh Laplace như dưới đây:
F(p)=6/p(2p^2+4p +10)
It looks like we're given the Laplace transform of f(t),
[tex]F(p) = L_p\left\{f(t)\right\} = \dfrac6{p(2p^2+4p+10)} = \dfrac3{p(p^2+2p+5)}[/tex]
Start by splitting up F(p) into partial fractions:
[tex]\dfrac3{p(p^2+2p+5)} = \dfrac ap + \dfrac{bp+c}{p^2+2p+5} \\\\ 3 = a(p^2+2p+5) + (bp+c)p \\\\ 3 = (a+b)p^2 + (2a+c)p + 5a \\\\ \implies \begin{cases}a+b=0 \\ 2a+c=0 \\ 5a=3\end{cases} \implies a=\dfrac35,b=-\dfrac35, c=-\dfrac65[/tex]
[tex]F(p) = \dfrac3{5p} - \dfrac{3p+6}{5(p^2+2p+5)}[/tex]
Complete the square in the denominator,
[tex]p^2+2p+5 = p^2+2p+1+4 = (p+1)^2+4[/tex]
and rewrite the numerator in terms of p + 1,
[tex]3p+6 = 3(p+1) + 3[/tex]
Then splitting up the second term gives
[tex]F(p) = \dfrac3{5p} - \dfrac{3(p+1)}{5((p+1)^2+4)} - \dfrac3{5((p+1)^2+4)}[/tex]
Now take the inverse transform:
[tex]L^{-1}_t\left\{F(p)\right\} = \dfrac35 L^{-1}_t\left\{\dfrac1p\right\} - \dfrac35 L^{-1}_t\left\{\dfrac{p+1}{(p+1)^2+2^2}\right\} - \dfrac3{5\times2} L^{-1}_t\left\{\dfrac2{(p+1)^2+2^2}\right\} \\\\ L^{-1}_t\left\{F(p)\right\} = \dfrac35 - \dfrac35 e^{-t} L^{-1}_t\left\{\dfrac p{p^2+2^2}\right\} - \dfrac3{10} e^{-t} L^{-1}_t\left\{\dfrac2{p^2+2^2}\right\} \\\\ \implies \boxed{f(t) = \dfrac35 - \dfrac35 e^{-t} \cos(2t) - \dfrac3{10} e^{-t} \sin(2t)}[/tex]
Triangle DEF has sides of length x, x+3, and x−1. What are all the possible types of DEF?
Triangle DEF is scalene
Must click thanks and mark brainliest
The triangle DEF will be a scalene triangle as all the sides of the triangle are unequal.
What is a scalene triangle?A scalene triangle is a type of triangle which have all the sides to be unequal and similarly, all the angles will also be unequal to each other.
Given that:-
Triangle DEF has sides of length x, x+3, and x−1it is given that all the sides of the triangle are x, x+3, and x−1 we can clearly see that for any value of x all the three sides will have different values. we can conclude from this that the triangle DEF is a scalene triangle.
Therefore triangle DEF will be a scalene triangle as all the sides of the triangle are unequal.
To know more about the scalene triangle follow
https://brainly.com/question/16589630
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Given FE=23.5, find BD.
Answer:
11.75
Step-by-step explanation:
The required triangle is attached below :
The triangle AFE has it's by the mid segment as BD ;as B is the mid-point of line EA ; and D is the mid-point of line FA ;
HENCE, The Length of the midsegment BD = 1/2FE
Hence, BD =. 1/2 * 23.5
BD = 23.5 / 2 = 11.75
A triangle has base of 7 1/8 feet and height 6 1/4 feet. Find the area of a triangle as a mixed number.
Answer: The area is 22 17/64.
Step-by-step explanation:
base = 7 1/8 = 57/8
height = 6 1/4 = 25/4
area = 1/2*b*h
= 1/2*57/8*25/4
= 1425/64
= 22 17/64
For rehab after an injury a patient walks 200m on the first day each day he will increase the amount walked by 100m. How many total kilometers will the patient have walked after 12 days
Answer:
3.3km
Step-by-step explanation:
200m on first day
Increase 200 by 100 = 300 (200+100)
From 2nd day to 11th day
300×11
3300m
If 1000m = 1km
3300m =?
3300/1000
3.3km
I hope it helps
Solve for x: 2x – 1] + 5 < 13 please send help
Answer:
Step-by-step explanation:
2|x-1|+5 < 13
2|x-1| < 8
|x-1| < 4
-4 < x-1 < 4
-3 < x < 5
Been stuck on this since yesterday !!?!?
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
I have uploaded a graph for you. The x axis is the number of years. The y axis is the salary multiplied by 1000. I should have made the multiplication factor 10000 but 1000 will do.
The 5 given points are plotted in red. The blue line is the function.
The function is y = 5x + 35. That means for every year you add 5 times the year onto the salary.
No years is 35000
1 year is 1 * 5000 + 35000
2 years is 2 * 5000 + 35000 = 45000
6 years is 5 * 5000 * 35000 = 65000
and so on.
The point you want is x = 12
12 years is 12 * 5000 + 35000 = 95000
Forgot the graph
Find TAN
Instructions: Find the value of the trigonometric ratio. Make
sure to simplify the fraction if needed.
please mark this answer as brainlist
HELPSSS PLSSSS I need help!!
Step-by-step explanation:
The perimeter of the rectangle is
[tex]P = 2(4x + 2x) = 12x[/tex]
The perimeter of the octagon is
[tex]P = 8(1.5x) = 12x[/tex]
So for x = 1, the perimeter of the rectangle, as well as the octagon, is 12 cm. For x = 2, its 24 cm. For x = 3, it's 36 and so on. So the pattern here is with each integer increase in x, the perimeter increases by 12 cm. Also that the perimeters of both shapes are equal.
Evaluate the expression (2²x² over xy³ )² for x = 4 and y == 2.
Answer:
4
Step-by-step explanation:
What is the volume of a cone below?
What is the measure of m?
Answer:
√245
Step-by-step explanation:
altitude on hypotenuse theorem:
m^2=7*35
m^2=245
m=√245
I need to know the answer please
Focusing on the center point of f(x) (0,0), we can see that it has moved to the left 4 units and up 3 units.
g(x) = [tex](\sqrt[3]{x + 4}) + 3[/tex]
Option C
Hope this helps!
plot the following points on a xy-plane.
(5,2) , (-2, 1) , (-1,-3)
Answer: See below
Step-by-step explanation:
Answer:
Answer below
Step-by-step explanation:
Which of the following can be used as "reasons" in a two-column proof?
Answer:
A definition and a theorem can be used as a reason in a two-column proof. A two column proof is assembled into statement and reason columns, where each statement should have verified reason.
Step-by-step explanation:
Answer:
Definitions and algebraic properties
Step-by-step explanation:
4
5
7
11
19
?
a. 41
b. 35
c. 23
d. 29
Answer:
35
Step-by-step explanation:
The pattern is adding powers of 2.
4+1=5 (exception)
5+2=7
7+4=11
11+8=19
19+16=35
Answer:
35
Step-by-step explanation:
4 + 1 = 5
5 + (1 × 2) = 5 +2 =7
7 + (2×2) = 7 + 4 = 11
11 +(4×2) = 11 + 8 = 19
19 + (8×2) = 19 + 16 = 35
Oscar has 1/5 of a jar of mustard. He puts equal amounts of the mustard onto 7 sandwiches and uses all of the mustard. What fraction of a jar of mustard does each sandwich have?
Answer:
1/35 jar of mustard yuh yuh
Solve the following formula for a.
Answer:
B is correct .trust me
Step-by-step explanation:
Help me with this question plz
9514 1404 393
Answer:
17
Step-by-step explanation:
The points at the ends of the interval are ...
(0, f(0)) = (0, 0)
(7, f(7)) = (7, 119)
The average rate of change is given by the slope formula:
m = (y2 -y1)/(x2 -x1)
m = (119 -0)/(7 -0) = 119/7 = 17
Will give brainliest answer
X+y=11
Graphing which function
Answer:
Step-by-step explanation:
slopee -1
y-intercept (0,11)
x y
0 11
1 10
Help me out!! Anyone
Answer:
4:10
Step-by-step explanation:
if they have to wait for plane B and it arrives every 10 mins then 4:10 is the anser
find the maximum number of children to whom 30 sweaters and 45 trousers can be equally divided. also how many sweaters and trousers will each get?
Answer:
five kids .each 6 sweaters and 9 trousers
Step-by-step explanation:
fThe number of patients treated at Dr. Frank's dentist office each day was recorded for nine days: 18, 19, 19, 4, 14, 8, 22, 3, 1. Using the given data, find the mean for this sample.
Answer:
12
Step-by-step explanation:
To find the mean, add up all the numbers
18+ 19+ 19+ 4+14+ 8+ 22+ 3+ 1
108
Then divide by the number of terms
108/9 =
12
State the domain and range of the following function:
{(- 3,4), (0,6), (2, - 2), (1, – 3), (6, - 7), (3, 2)}
Answer:
[tex]Domain = \{-3,0,2,1,6,3\}[/tex]
[tex]Range = \{4,6,-2,-3,-7,2\}[/tex]
Step-by-step explanation:
Given
[tex]\{(- 3,4), (0,6), (2, - 2), (1, - 3), (6, - 7), (3, 2)\}[/tex]
Required
The domain and range
A function is represented as:
[tex]Function = \{(x_1,y_1),(x_2,y_2),....(x_n,y_n)\}[/tex]
Where
[tex]x \to[/tex] domain
[tex]y \to[/tex] range
So, we have:
[tex]Domain = \{-3,0,2,1,6,3\}[/tex]
[tex]Range = \{4,6,-2,-3,-7,2\}[/tex]
an expression equivalent to 12+ 21 is
Answer:
33
Step-by-step explanation:
The answer is 1628. Because 16 x 21 = 28 x 12 = 336
Hope this helps
