1 8. 1 (minutes) 0 5 6 g(t) (cubic feet per minute) 12.8 15.1 20.5 18.3 22.7 Grain is being added to a silo. At time t = 0, the silo is empty. The rate at which grain is being added is modeled by the differentiable function g, where g(t) is measured in cubic feet per minute for 0 st 58 minutes. Selected values of g(t) are given in the table above. a. Using the data in the table, approximate g'(3). Using correct units, interpret the meaning of g'(3) in the context of this problem. b. Write an integral expression that represents the total amount of grain added to the silo from time t=0 to time t = 8. Use a right Riemann sum with the four subintervals indicated by the data in the table to approximate the integral. πί c. The grain in the silo is spoiling at a rate modeled by w(t)=32 sin where wſt) is measured in 74 cubic feet per minute for 0 st 58 minutes. Using the result from part (b), approximate the amount of unspoiled grain remaining in the silo at time t = 8. d. Based on the model in part (c), is the amount of unspoiled grain in the silo increasing or decreasing at time t = 6? Show the work that leads to your answer. =
Answers
Hi there!
a)
We can approximate g'(3) by finding the slope of g(t) over an interval containing t = 3.
We can use the endpoints t = 1 and t = 5 min for the best estimate.
[tex]slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{20.5-15.1}{5-1} = \boxed{1.35 \frac{ft^3}{min^2}}[/tex]
This means that the rate of grain being added to the silo is INCREASING at a rate of 1.35 ft³/min². (Or in other words, the grain is being poured at an increasingly greater rate)
b)
The total amount of grain added is the integral of g(t), so:
[tex]\text{total amount of grain} = \int\limits^8_0 {g(t)} \, dt[/tex]
We can do a right riemann sum by using the right endpoints (t = 1, t = 5, t = 6, t = 8) to calculate.
Riemann sums are essentially rectangles added up to calculate an approximate value for the area under a curve.
The bases are the spaces between each value in the chart, while the heights are the values of g(t).
Using the intervals and values in the chart:
[tex]1(15.1) + 4(20.5) + 1(18.3) + 2(22.7) = \boxed{160.8 ft^3}[/tex]
c)
We can subtract the two integrals to find the total amount of unspoiled grain.
With g(t) being fresh grain and w(t) being spoiled grain, let y(t) represent unspoiled grain.
[tex]y(t) = \int\limits^8_0 {g(t)} \, dt - \int\limits^8_0 {w(t)} \, dt[/tex]
Use a calculator to evaluate:
[tex]y(t) = 160.8 - \int\limits^8_0 {w(t)} \, dt = 160.8 - 99.05 = \boxed{61.749 ft^3}[/tex]
d)
We can do the first derivative test to determine whether the amount of grain is increasing or decreasing. (Whether the first derivative is positive or negative at this value).
For the above integral, we know that the derivative is:
[tex]y'(t) = g(t) - w(t)[/tex]
Plug in the values for t = 6:
[tex]w(6) = 32 \cdot \sqrt{sin(\frac{\pi (6)}{74})} = 16.06[/tex]
[tex]y'(6) = g(6) - w(6) = 18.3 - 16.06 = 2.23 \frac{ft^3}{min}[/tex]
This value is POSITIVE, so the amount of unspoiled grain is INCREASING.
Related Questions
NO LINKS!!! Draw and solve each triangle.
1. a= 3, b= 4, C= 40°
2. a= 3, c= 2, B=110°
3. a= 12, b= 13, , c= 5
4. a= 5, b= 8, c = 9
Answers
The missing values are highlighted in bold, which are approximate to four decimal places (with the exception of B = 90 in problem 3).
Lowercase letters a,b,c are side lengths; Uppercase letters A,B,C are angles opposite the sides mentioned.
The drawings are shown below.
==========================================================
Explanation:
Problem 1
Use the Law of Cosines to solve for side c
c^2 = a^2 + b^2 - 2*a*b*cos(C)
c^2 = 3^2 + 4^2 - 2*3*4*cos(40)
c^2 = 6.61493337
c = sqrt(6.61493337)
c = 2.57195128
c = 2.5720
Now apply the Law of Sines to find angle A
sin(A)/a = sin(C)/c
sin(A)/3 = sin(40)/2.57195128
sin(A) = 3*sin(40)/2.57195128
sin(A) = 0.74976647
A = arcsin(0.74976647)
A = 48.57015285
A = 48.5702
Once we know two angles of a triangle, we can then subtract their sum from 180 to get the third missing angle.
A+B+C = 180
B = 180 - (A+C)
B = 180 - (48.57015285+40)
B = 91.42984715
B = 91.4298
-----------------------------------
Problem 2
Like with the previous problem, we can use the Law of Cosines to solve for the missing side.
b^2 = a^2 + c^2 - 2*a*c*cos(B)
b^2 = 3^2 + 2^2 - 2*3*2*cos(110)
b^2 = 17.10424172
b = sqrt(17.10424172)
b = 4.13572747
b = 4.1357
Now use the Law of Sines to find angle A
sin(A)/a = sin(B)/b
sin(A)/3 = sin(110)/4.13572747
sin(A) = 3*sin(110)/4.13572747
sin(A) = 0.68164014
A = arcsin(0.68164014)
A = 42.9719427
A = 42.9719
Then we can wrap things up like so
A+B+C = 180
C = 180 - (A+B)
C = 180 - (42.9719427+110)
C = 27.0280573
C = 27.0281
-----------------------------------
Problem 3
Use the Law of Cosines to find angle A.
a^2 = b^2 + c^2 - 2*b*c*cos(A)
12^2 = 13^2 + 5^2 - 2*13*5*cos(A)
144 = 194 - 130*cos(A)
-130*cos(A) = 144 - 194
-130*cos(A) = -50
cos(A) = (-50)/(- 130)
cos(A) = 0.38461538
A = arccos(0.38461538)
A = 67.38013505
A = 67.3801
We could use the Law of Cosines to find angle B, but I'll use the Law of Sines instead.
sin(A)/a = sin(B)/b
sin(B) = b*sin(A)/a
sin(B) = 13*sin(67.38013505)/12
sin(B) = 0.99999999998574
sin(B) = 1
B = arcsin(1)
B = 90
It turns out we have a 5-12-13 right triangle (though usually c is the longest side instead of b).
Now we could use the Law of Cosines to find angle C, but the steps shown below are much quicker in my opinion that don't involve any trig functions.
A+B+C = 180
C = 180 - (A+B)
C = 180 - (67.38013505 + 90)
C = 22.61986495
C = 22.6199
-----------------------------------
Problem 4
The steps will basically be identical to problem 3, but with different numbers of course.
First we need the Law of Cosines to find angle A.
a^2 = b^2 + c^2 - 2*b*c*cos(A)
5^2 = 8^2 + 9^2 - 2*8*9*cos(A)
25 = 145 - 144*cos(A)
-144*cos(A) = 25 - 145
-144*cos(A) = -120
cos(A) = (-120)/(-144)
cos(A) = 0.83333333
A = arccos(0.83333333)
A = 33.55730976
A = 33.5573
Now use the Law of Sines to find angle B.
sin(A)/a = sin(B)/b
sin(B) = b*sin(A)/a
sin(B) = 8*sin(33.55730976)/5
sin(B) = 0.88443328
B = arcsin(0.88443328)
B = 62.18186103
B = 62.1819
Lastly,
A+B+C = 180
C = 180 - (A+B)
C = 180 - (33.55730976 + 62.18186103)
C = 84.26082921
C = 84.2608
-----------------------------------
Side notes:
The phrasing "solve the triangle" means to find all three sides and all three angles.Each answer is approximate with the exception of B = 90 in problem 3.Intermediate calculations were done to eight decimal places. The final answers are to four decimal places. In the first two problems, we can prove only one triangle is possible due to the SAS triangle congruence theorem.For the last two problems, only one triangle is possible because of the SSS triangle congruence theorem.I used GeoGebra to make each triangle shown below.Look at VSW and ZTSW in the image below.
Which of the following is the best description for this pair of angles?
O complementary
O supplementary
O acute
O straight
Answers
Answer:
Supplementary
Step-by-step explanation:
VSW and TSW combine and make 180. Complementary is when they make up to 90
Supplementary
What is the answer for X+8
Answers
Answer: the answer is x+8= 1
Determine if true:
3x5+8=23
Answers
Answer:
True
Step-by-step explanation:
3x5=15
15+8=23
23=23
Answer:
True
Step-by-step explanation:
3 x 5 = 15
15 + 8 = 23
for function f(x) =-3x+3 find f(0).
Answers
Answer:
f(0)=3
Step-by-step explanation:
f(0) = -3 x 0 + 3
f(0)=3
out of 30 students in a class,5 said they ride there bikes to school. based on these relusts, how many of the 600 students ride there bike to school
Answers
Answer:
125 students ride their bike to school
Step-by-step explanation:
We can set a proportion.
5 / 32 = x / 800
where x is the number of students out of 100 who ride their bikes.
Cross multiply.
32x = 4000
Divide both sides of the equation by 32.
x = 125
125 students out of 800 students ride their bikes to school.
at a stationery store 6 pens were sold at $20. Mr. wong bought pens for his 36 students. each student got 2 pens. how much did he spend altogether?
Answers
Answer:
[tex](20 \div 6) \times (36 \times 2) = 3.30 \times 72 = 237.60[/tex]
HOPE IT HELP!!A bag of marbles contains five red marbles three green marbles and six yellow marbles what is the probability of randomly choosing a marble that is green
Answers
Numbers x and y both squared and added
And ASAP pls
Nira here :)
Answers
Answer:
2x² + 4y²
according to your given information.......
this season the baseball team won 3/5 of its games. assuming there’s no ties, what is the ratio of games won to games lost
Answers
Answer:
3:2
Step-by-step explanation:
if they won 3/5 games, then the other 2 were lost so the ratio would be 3:2
p multiple by 4 written as a
algebraic expression
Answers
Answer:
4p
Step-by-step explanation:
you dont have to put multiplication sign when you multiply by variables this is honestly a pretty easy question
6. How do you write 100 as a power of 10?
Answers
Answer:10 to the 2nd power
Step-by-step explanation:
3 ounces cost $273
Find the unit rate
Answers
Answer:
$91 per ounce
Step-by-step explanation:
If 3 ounces cost $273, we can divide both sides by 3 to find the unit rate. To make this clearer, we use the variable [tex]x[/tex] for ounces. Then, we have the equation
[tex]3x=273[/tex].
Dividing both sides by 3, we have
[tex]x=91[/tex].
The cost of one ounce is $91.
2 ft is equal to how many meters?
Answers
Answer:
2 feet is equal to 0.609 meters
One meter is equal to roughly 3.281 feet.
So two feet is less than one meter.
To find how many meters a certain amount of feet is, you divide by 3.281
2 ÷ 3.281 = 0.609
2 feet is equal to 0.609 meters
Answer:
0.61 meter
Step-by-step explanation:
Urgent need help pls answer with correct explanation
Answers
Answer:
See below.Step-by-step explanation:
Question 1:
A beach ball is a sphere. It's that simple.Question 2:
This requires your eyes to find it. On the right-middle side, there is a prism that says 3, 2, and 4. The order to multiplying this is 3 * 2 * 4. The answer is 24.Question 3:
Can't solve this sorry. :(
At a party there are 40 prizes, which are either kazoos or whistles. The tape diagram shows the ratio of kazoos to whistles.
Complete the table.
Original ratio Prizes at the party
Kazoos 3 : -
Whistles 5 : -
Total prizes - :40
Answers
Out of the 40 prizes, 15 prizes were for Kazoos and 35 prizes were for whistle
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given the diagram, the ratio is represented, hence:
Number of Kazoos = (3/8) * 40 = 15
Number of whistles = (5/8) * 40 = 35
Out of the 40 prizes, 15 prizes were for Kazoos and 35 prizes were for whistle
Find out more on equation at: https://brainly.com/question/2972832
zearn lesson really hard
Answers
Answer:7541
Step-by-step explanation:
all you have to do is add the bottom
Pls help me solve part a and part b
Answers
Find the radius of the cylinder with the given volume
WILL GIVE BRAINLIEST
Answers
Answer:
6.71 mm
Step-by-step explanation:
Find radius R
Volume = πR²H
4500π = πR²*100
R² = 45
so R = √45 = 3√5 = 6.71 mm
Which is the product of 7/9 and 6?
O A. 1 4/9
о в. 3 7/9
O c. 4 2 /3
o D. 7 5/7
Answers
pls help me find the awnser
Answers
Answer:
I have no clue thats why is got this app
Answer:
57 units³
Step-by-step explanation:
Volume = length • width • height
Fill in the numbers:
V = 3 4/5 • 2 1/2 • 6
I am making them into improper fractions to make it easier:
V = 19/5 • 5/2 • 6
Multiplying the length and the width:
V = 95/10 • 6
Multiplying the height in:
V = 57
How many solutions can be found for the system of linear equations represented on the graph?
A) no solution
B) one solution
C) two solutions
D) infinitely many solutions
Answers
Answer:
Infinetly many (d)
Step-by-step explanation:
The graphs are the same, so they will all have the same solutions.
hope this helps!
ibrahim
Easy word problem (click photo)
Answers
Answer:
90 apples
Step-by-step explanation:
1st class did 1/3, 1/3 of 60 is 20, 60-20 is 40, 40 into 2.25 or 2 1/4 = 90 because 40 x 2 = 80 and 1/4 of 40 is 10, 10+80 = 90, so the second class needs 90 apples
Answer:
90 apples
Step-by-step explanation:
First, we need to find 1/3 of the total number of pies, which is the number of pies the first class made. To do this, we divide 60 by 3 (or multiply 60 by 1/3):
60 ÷ 3 = 20
This means the first class made 20 pies and that the second class has to make 40 pies (60 - 20 = 40)
1 pie needs 2 1/4 apples
To find how many apples the second class will need, we must multiply 40 (the number of pies) • 2 1/4 (the number of apples needed for 1 pie)
40 • 2 1/4 = 90
This means 90 apples are needed for the second class to make the remaining pies
Hey guys, I need your help!
The whole permutation stuff is pretty new to me. My teacher taught the class this recently, but I am still unsure on what to do. I would love it if anyone out there who can provide a step-by-step instructions and an explanation please!
Answers
Step-by-step explanation:
Permutation is "How many ways can you arrange 'r' from a set of 'n' if the order matters?"
For #7, it would be "how many ways can you pick 2 things out of 6 things when the order matters?"
The steps for this is in the picture.
For #9, you would do the same as #7 and just add -4 to your final answer
What % of people surveyed
were 6-10 and 21-25 years old
Answers
Add total people:
5 + 20 + 40 + 30 + 20 = 115 people
Add people 6-10 to 21-25 = 20 + 20 = 40
Divide the two age ranges by total people:
40/115 = 0.34782609
Multiply by 100 to get percent:
0.34782609 x 100 = 34.782609% Round off as needed
Solve for x .
A) 48
B)42
C)47
D) 44
Answers
Answer:
B. 42.
Step-by-step explanation:
x/35 = 30/25
25x = 30*35
x = (30*35)/25
= 1050/25
= 42.
Factor the greatest common factor (GCF) from each polynomial. 81ab-90a
Answers
Answer:
9a(9b-10)
Step-by-step explanation:
1 All sides are congruent. Paralelogram Rhombus Rectangle Square
Answers
Answer:
Rhombus
Square
Step-by-step explanation:
Rhombuses and squares have sides that are always congruent.
choose the sentence that complete the fact family like 3×6=18 18÷6=3 6×3=18 is it a 6÷3=2 b 18÷3=6 c 9÷3=3 d 18÷9=2 is it abc or d
Answers
Answer:
b 18÷3=6
Step-by-step explanation:
The fact family contains 3,6and 18
since 3×6=18,18÷6=3and 6×3=18
∴ you need to look for numbers that contain 3,6and18. when you find the number make sure to check that they are not in the question, if it is not in the question then that is your answer.
hope it helps
Change 288 square inches to feet
Answers
[tex]______________________________________________[/tex]
[tex]\huge\rm{ANSWER:}[/tex]
2 square feet[tex]\huge\rm{GIVEN:}[/tex]
Change 288 square inches to feet
(Keep in mind that 1 feet equals 12 inches)
[tex]\huge\rm{SOLUTION:}[/tex]
Follow these steps to convert to square feet.
Step 1: Convert an inch to a square inch.
If 1 feet equals 12 inches, then
1 square feet equals (12) inches equals 144 square inches
As a result, 144 square inches equals one square feet.
Step 2: Using the unitary method, we obtain
288 square inches = 288/144 square feet.
288 square inches = 2 square feet
As a result, 288 square inches equals 2 square feet.
[tex]______________________________________________[/tex]