The critical numbers of the function f(θ) = 8sec(θ) + 4tan(θ) are where the derivative is undefined, specifically at θ = π/2 and θ = 3π/2.
Explanation:To find the critical numbers of the function f(θ) = 8sec(θ) + 4tan(θ), we need to take its derivative and set it equal to zero or find where the derivative does not exist. First, we compute the derivative of the function. Recall that the derivative of sec(θ) is sec(θ)tan(θ) and the derivative of tan(θ) is sec2(θ).
Thus, the derivative f'(θ) is 8sec(θ)tan(θ) + 4sec2(θ). We look for where this derivative is undefined or equal to zero. The derivative is undefined when sec(θ) is undefined, which occurs at θ = π/2 and θ = 3π/2. To find where the derivative is zero, we can factor out 4sec(θ):
4sec(θ)(2tan(θ) + sec(θ)) = 0
The second factor will never be zero because sec(θ) and tan(θ) will not be zero for the same θ in the interval given. Therefore, the critical numbers are where the derivative is undefined, which is θ = π/2 and θ = 3π/2.
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Which sentence explains the correct first step in the solution of this equation?
4(x−3)=9
A. Add 2 to both sides.
B. Apply the distributive property to get 4x - 12 = 9.
C. Apply the distributive property to get 4x - 3 = 9.
D. Subtract 9 from both sides.
Answer:
b
Step-by-step explanation:
5/8 divided by 5 3/4
six people share 3/5 pound of peanuts equally. What fraction of a pound of peanuts does each person recieve?
which expression represents the perimeter of a rectangle above? the width is (3y-5)ft and the length is 2yft
a. (5y-5)
b. (10y+10)
c. (10y-10
d. (6y^2-10y)
The perimeter of a rectangle is 10(y-1).
Option (c) is correct.
It is required to find the expression of perimeter of a rectangle.
What is rectangle?A rectangle is a type of quadrilateral that has its parallel sides equal to each other and all the four vertices are equal to 90 degrees.
Given that:
Let the length of rectangle be 3y-5
Let the breadth of rectangle be 2y
As we know the formula for perimeter,
So,
Perimeter (P)= 2Length (L) +2Width (W)
P=2L + 2W
P=2(2y)+2(3y-5)
P=4y+6y-10
P=10y-10
∴P=10(y-1)
Therefore, the perimeter of a rectangle is 10(y-1).
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Write a number sentence to show the distributive property. 7x6=
Answer:
According to the distributive property of multiplication,
a . (b + c ) = (a . b) + (a . c)
Given: 7 × 6
which can be rewritten as: a) 7 . (4 + 2), b) 7 . (3 + 3), c) 7 . (1 + 5)
Rewriting the given expression using the distributive property of multiplication:
a) 7 . (4 + 2) = (7 . 4) + (7 . 2)
7 . (6) = 28 + 14
42 = 42
b) 7 . (3 + 3) = (7 . 3) + (7 . 3)
7 . (6) = 21 + 21
42 = 42
c) 7 . (1 + 5) = (7 . 1) + (7 . 5)
7 . (6) = 7 + 35
42 = 42
As the left hand side is equal to the right hand side in each case, therefore, it proves the distributive property.
Using the information from problem #1, if a 32 inch Northern Pike is caught, then the weight in pounds as predicted by the least-squares line is ____ pounds. (Round your answer to 2 decimal places).
Given the points L(-2,5) and M(2,-3), point Q (6/5 m, -7/5) partitions LM in the ratio
Answer:
4:1
Step-by-step explanation:
We are given that
Point L(-2,5) and M(2,-3).
Point Q([tex]\frac{6}{5},\frac{-7}{5})[/tex] partitions LM .
We have to find the ratio in which Q divides the LM.
Let point Q divides the LM in the ratio K:1.
Section formula;[tex]x=\frac{m_1x_2+m_2x_1}{m_1+m_2},y=\frac{m_1y_2+m_2y_1}{m_1+m_2}[/tex]
Substitute the values in the given formula then we get
[tex]\frac{6}{5}=\frac{2k-2}{k+1}[/tex]
[tex]6k+6=10k-10[/tex]
[tex]6+10=10k-6k[/tex]
[tex]16=4k[/tex]
[tex]k=\frac{16}{4}=4[/tex]
Hence, the point Q divides the LM in the ratio 4:1.
Yolanda's club has 50 members. It's rules require that 60% of them must be present for any vote. At least how many members must be present to vote?
Daniel dives into a swimming pool. The height of his head is represented by the function h(t) = -8t2 - 28t + 60, where t is the time in seconds and h(t) is the height of his head, in inches.
How many seconds will it take for Daniel's head to reach the surface of the pool?
Write each percent as a fraction in simplest form 47%=
The percentage of 47% can be expressed as the fraction 47/100 in simplest form. This means that 47 out of every 100 units can be represented by the fraction 47/100.
The percentage is given as 47%.
To write 47% as a fraction in simplest form, we can interpret the percent symbol as meaning "per 100."
Recognize that percent means "per 100." Therefore, 47% can be written as 47 per 100 or 47/100.
Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. In this case, the GCD of 47 and 100 is 1.
Divide both the numerator and denominator by their GCD to simplify the fraction:
(47 ÷ 1) / (100 ÷ 1) = 47/100
The resulting fraction, 47/100, is already in its simplest form.
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Carly is 5 years younger then Lisa. Lisa is 3 times as old as Mark. What is a algebraic expression to represent Carly's age, in years if mark is m years old explain
Mark = m
Lisa = 3m ( 3 times as old as Mark)
Carly = 3m-5 ( 5 years younger than Lisa)
expression is C = 3m-5
Twelve of the students in the school choir like to sing solos. These 12 students make up 24% of the choir. How many students are in the choir?
Answer:
The answer is 50 students
Step-by-step explanation:
solve 8y-3(4-2y)=6(y+1)
Malia has 8 hamsters. That is 6 fewer than Sasha has. How many hamsters does Sasha have?
A 12 oz bottle of a new soda costs $2.49. What is the unit rate, rounded to the nearest tenth of a cent?
there are 7 days in 1 week. some months consist of 30 days. how manu weeks are there in 280 days
There are a total of 40 weeks in 280 days.
What is Division?Division is one of the four fundamental arithmetic operations, which tells us how the numbers are combined to form a new one.
Given that there are 7 days in 1 week and some months consist of 30 days. Now, the number of weeks that 280 days will have can be found as,
Number of weeks = Total number of days / Number of days in a week
= 280 days / 7 days per week
= 40 week
Therefore, There are 40 weeks in 280 days.
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In a suvey of 100 out-patients who reported at a hospital one day it was find out that 70 complained of fever, 50 complained of stomachs and 30 were injured. All 100 patients had at least one of the complains. How many patients had all three complains?
Final answer:
To find the number of patients who had all three complaints (fever, stomach issues, and injury), we can use a Venn diagram. From the given information, we determine that 30 patients had all three complaints.
Explanation:
To find the number of patients who had all three complaints, we can use a Venn diagram. Let's start with the information given:
Total number of patients = 100Number of patients who complained of fever = 70Number of patients who complained of stomach issues = 50Number of patients who were injured = 30All 100 patients had at least one complaintTo determine the number of patients who had all three complaints, we need to find the overlapping region in the Venn diagram. From the given information, we can determine the following:
The number of patients who complained of fever and stomach issues = 100 - 30 (injured patients) = 70The number of patients who complained of fever and were injured = 70 - 50 (patients with stomach issues) = 20The number of patients who complained of stomach issues and were injured = 50 - 20 (patients with both fever and injury) = 30The number of patients who had all three complaints = 30 (those who complained of stomach issues and were injured)Therefore, the number of patients who had all three complaints is 30.
Final answer:
To find out how many patients had all three complaints of fever, stomach pain, and injury, we used the principle of inclusion-exclusion and found that 25 patients had all three complaints.
Explanation:
To determine how many patients had all three complaints (fever, stomach pain, and injury) at the hospital, we can use the principle of inclusion-exclusion from combinatorics.
First, we add the number of people with each complaint:
People with fever: 70
People with stomach pain: 50
People with injury: 30
According to the inclusion-exclusion principle:
Total with all complaints = (People with fever) + (People with stomach pain) + (People with injury) - (People with fever and stomach pain) - (People with fever and injury) - (People with stomach pain and injury) + (People with all three complaints)
Since all 100 patients had at least one complaint, and assuming the least number of people with multiple complaints, we minimize the terms (People with fever and stomach pain), (People with fever and injury), and (People with stomach pain and injury). The minimum number for all these combined categories is the sum of individual complaints minus the total number of patients, which is (70 + 50 + 30) - 100 = 50.
Therefore, if there were no people with all three complaints, the total with any two complaints would be 50. However, this number includes people with all three complaints three times (once for each pair of complaints). To correct this, we subtract twice the number of people with all three complaints to get the true number of people with exactly two complaints:
50 - 2*(People with all three complaints) = Total with exactly two complaints
If there are no people with exactly two complaints, then this means that all 50 are those with all three complaints:
50 - 2*(People with all three complaints) = 0
Solving for (People with all three complaints), we get:
(People with all three complaints) = 50 / 2
(People with all three complaints) = 25
This result suggests that 25 patients had all three complaints of fever, stomach pain, and injury.
which comparison sentence best represents the equation 6 x 7=42? 7 is 6 times as many as 42? or 6 is 7 times as many as 42 or 42 is 6 times as many as 6 or 6 more than 7 is 42
The comparison sentence that best represents the equation 6 x 7 = 42 is option 3) 42 is 6 times as many as 7.
Let's break it down step-by-step:
Identify the numbers: 6, 7, and 42.Determine the relationship established by the equation: Multiplication.Apply the multiplication: 6 multiplied by 7 equals 42.Formulate the sentence based on this relationship: '42 is 6 times as many as 7.'Therefore, the correct choice is option 3) 42 is 6 times as many as 7.
Stefan is conducting an experiment to find the level of pollution in the air. He found that there are about 3 billion dust particles in one square meter of space. What is the approximate weight of 3 billion dust particles, if one dust particle weighs 4.66 × 10-12 grams? 1.4 × 10-4 grams
What is the sum of 1/3(9-6m) + 1/4(12m -8)?
Answer:
The sum of the given expression is (m+1).
Step-by-step explanation:
The given expression is
[tex]\frac{1}{3}(9-6m)+\frac{1}{4}(12m-8)[/tex]
Use distributive property to to simplify the given expression.
[tex]\frac{1}{3}(9-6m)+\frac{1}{4}(12m-8)=\frac{1}{3}(9)+\frac{1}{3}(-6m)+\frac{1}{4}(12m)+\frac{1}{4}(-8)[/tex]
[tex]\frac{1}{3}(9-6m)+\frac{1}{4}(12m-8)=3-2m+3m-2[/tex]
Add like terms.
[tex]\frac{1}{3}(9-6m)+\frac{1}{4}(12m-8)=(3-2)+(-2m+3m)[/tex]
[tex]\frac{1}{3}(9-6m)+\frac{1}{4}(12m-8)=(1)+(m)[/tex]
[tex]\frac{1}{3}(9-6m)+\frac{1}{4}(12m-8)=m+1[/tex]
The sum of the given expression is (m+1).
7j+5-8k
when j=0.5 and k=0.25
Answer:
6.5
Step-by-step explanation:
To perform this kind of exercise, it is necessary to replace the value indicated in each variable each time you find the variable in the equation.
[tex]7j+5-8k[/tex]
If j=0.5 and k=0.25, replace the value of each variable.
[tex]7(0.5)+5-8(0.25)=[/tex]
now perform the indicated operations
[tex]7(0.5)+5-8(0.25)=3.5+5-2=6.5[/tex]
Answer:
6.5
Step-by-step explanation:
4 1/2 + 8 3/4
A. 12 2/3
B. 12 1/3
C. 13 1/4
D. 13 2/3
4 1/2 + 8 3/4
4+8 = 12
1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 1/4
12 + 1 1/4 = 13 1/4
Answer is C
The intelligence quotient (iq) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. what is the probability we could select a sample of 50 adults and find that the mean of this sample is between 98 and 103?
Photography World has a portrait special. A three-pose portrait package has a sitting fee of $35, and a six-pose package has a sitting fee of $60. On a Saturday, $690 was collected in sitting fees. Which equation can be used to represent x, the number of three-pose portrait packages and y, the number of six-pose portrait packages?
The equation that can be used to represent the number of three-pose and six-pose portrait packages is: 35x + 60y = 690
What are word problems?Word problems in mathematics are methods and approaches applied in solving real-life scenarios. Data can be modeled from the certain information given.
From the parameters given, Let us assume that:
x represents a three-pose portrait package that has a sitting fee of $35y represents a six-pose package that has a sitting fee of $60Now, on Saturday a total amount of 690 was collected. This data can be modeled as:
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In isosceles ∆ABC the segment BD (with D ∈ AC ) is the median to the base AC . Find BD, if the perimeter of ∆ABC is 50m, and the perimeter of ∆ABD is 40m.
The length of the height triangle ABC is BD i.e., [tex]15\;units[/tex].
According to the question, In isosceles [tex]∆ABC[/tex] the segment BD is the median to the base AC . Also, the perimeter of ∆ABC is 50m, and the perimeter of the triangle [tex]ABD[/tex] is 40m.
Let the length of the equal sides of the isosceles triangle be [tex]x \;meters[/tex] and the length of the base of the isosceles triangle is [tex]2y\;meters[/tex].
[tex]Perimeter_{ABC}=x+x+2y\\2x+2y=50\\x+y=25\\x=25-y----(i)[/tex]
Also,
[tex]Perimeter_{ABD}=x+y+BD\\x+y+\sqrt{x^2-y^2}=40\\x+y+\sqrt{(25-y)^2-y^2}=40\\25+\sqrt{625-50y}=40\\\sqrt{625-50y}=15\\625-50y=225\\y=8\;units[/tex]
So,
[tex]x=25-8\\x=17\;units[/tex]
Now,
[tex]BD=40-x-y\\BD=40-17-8\\BD=15\;units[/tex]
Hence, the length of the height triangle ABC is BD i.e., [tex]15\;units[/tex].
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28 is 32% of what number?
divide:
28 / 0.32 = 87.5
an airport offers three shuttles that run on different schedules. if all shuttles leave the airport at 4:00P.M., at what time will they next leave the airport together? a leaves every 8 mins, b every 10 mins and c every 12 mins. When is the next time they will all leave together
The height of a grain of a cylindrical silo is is increasing at a constant rate of 4 feet per minute At what rate is the volume of grain in the cylinder if the radius of the silo is 10 feet?
Just need the set up of the derivative
convert 0.045 to a fraction
Whats the unit rate of $3.60