Using the mean value theorem, the least possible value of f(3) is - 2
What is the mean value theorem?The mean value theorem states that for a function f(x) on the interval (a, b), there exists a value c such that
f'(c) = f(b) - f(a)/(b - a)
How to find the least possible value for f(3)?Since ƒ(−1) = 10 and ƒ'(x) ≥−3 for all values of x, we require the least possible value of f(3).
Using the mean value theorem where
b = 3, a = -1 and f'(c) ≥ 3.We choose f'(c) = 3 since f'(x) = 3 is the minimum value of f'(x).
So, substituting the values of the variables into the equation, we have that
f'(c) = f(b) - f(a)/(b - a)
f'(x) = f(3) - f(-1)/(3 - (-1))
f'(x) = f(3) - f(-1)/(3 + 1)
f'(x) = f(3) - f(-1)/4
So, making f(3) subject of the formula, we have that
f(3) = f(-1) + 4f'(x)
Since
f'(x) = 3 and f(-1) = 10,Substituting the values of the variables into the equation, we have that
f(3) = f(-1) + 4f'(x)
f(3) = 10 + 4(-3)
f(3) = 10 - 12
f(3) = - 2
So, the least possible value is - 2
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Suppose that 15 inches of wire costs 90 cents. At the same rate, how many inches of wire can be bought for 72 cents?
Answer:
12 inches-----------------------------
15 inches of wire costs 90 cents, then 1 inch of wire costs:
90/15 = 6 centsThe length of wire for 72 cents:
72/6 = 12 inchesThe scale model of a statue is 1 inch to 11.2 feet. If the model is 5.8 inches, what is the length, in feet, of the actual statue?
The length of the actual statue in feet would be = 64.96 ft
What is scale model?A scale model is defined as the the exact type of an object but with a different measurement.
The scale model of a statue = 1 In = 11.2 ft
The length of the model = 5.8 inches
Therefore it length in feet = X ft
That is; 1 in = 11.2 ft
5.8 in = X ft
make X ft the subject of formula;
X ft = 5.8 × 11.2 = 64.96 ft
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(Help within 5 minutes)
A small town surveyed 250 adults, 180 of whom were parents, about whether a park should be expanded. The results showed that 150 of the adults who were parents and 40 of the adults who were not parents thought the park should be expanded.
Use this information to complete the two-way table.
Enter your answer by filling in the boxes to complete the table.
Should the park be expanded?
Yes No
Parents
Not parents
The park should be expanded based on the information as the majority want the expansion.
How to illustrate the information?From the information, on the small town surveyed 250 adults, 180 of whom were parents, about whether a park should be expanded.
Also, the results showed that 150 of the adults who were parents and 40 of the adults who were not parents thought the park should be expanded.
Based on the above the number of people that wants the expansion will be:
= (150 + 40) / 250 × 100
= 190 / 250 × 100
= 76%
It should be expanded.
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A business has $11,080 to spend on new laptops and tablet computers for its salespeople. The laptops cost $515 each. The tablets cost $285 each. The business wants each salesperson to have either a laptop or a tablet. There are 30 salespeople. Create a system of equations that models how many of each type of computer the business should buy. 515x 285y
Answer:
515 x + 285 y = 11080
x + y = 30
Step-by-step explanation:
I need help with this question please
The average rate of change is the ratio of the difference between the y coordinates of two points to the x coordinate of the same point.
It can be found by using two points on the graph and substitute into the formula: (y₀ - y₁) / (x₀ - x₁)
The square brackets [] is used to mean the end points as shown is inclusive
What is average rate of change in quadratic equation?A quadratic equation is a polynomial of degree two
The average rate of change here is not constant and is usually specified by the two points to which it is measured
For instance using points (-3, 0) and (0, -1) we have
= (y₀ - y₁) / (x₀ - x₁)
= (-1 - 0) / (0 - (-3))
= -1/3
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Answer:
7/12
Step-by-step explanation:
y = 12
x = 7
y = Kx
K = y/x
K = 7/12
What is the largest prime factor of 999?
Answer:
37
Hope this helps :)
According to the direct instruction, what is the recommended amount of servings for milk or milk products per day?
a.5 cups
b.3 cups
c.1 cup
Three glasses of milk or dairy products per day are recommended for adults. One serving of milk is one cup. According to the USDA, a serving of milk-based goods is the amount that corresponds to one cup of milk.
What are dairy products ?Food items manufactured from (or containing) milk are referred to as dairy products or milk products (also known as lacticinia).
The cow, water buffalo, nanny goat, and ewe are the most popular dairy animals.
Dairy products include staples from Western grocery stores including yogurt, cheese, and butter.
A dairy is a place where dairy products are made .Various amounts of dairy products are consumed worldwide (see consumption patterns worldwide) .
After standardizing the fat content and optional homogenization or pasteurization, milk is produced in a variety of grades with the potential addition of the microorganisms Streptococcus lactis and Leuconostoc citrovorum.
Based on the type of product generated, milk can be divided into a number of distinct categories, including cream, butter, cheese, newborn formula, and yogurt.
Hence, Three glasses of milk or dairy products per day are recommended for adults. One serving of milk is one cup. According to the USDA, a serving of milk-based goods is the amount that corresponds to one cup of milk.
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Using traditional methods it takes 8.5 hours to receive a basic flying license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique on 21 students and observed that they had a mean of 8.8 hours with a variance of 1.21 . Is there evidence at the 0.05 level that the technique performs differently than the traditional method? Assume the population distribution is approximately normal. Step 1 of 5 : State the null and alternative hypotheses
As the p-value of the test is of 0.215 > 0.05, there is not enough evidence at the 0.05 level that the technique performs differently than the traditional method.
What are the null and the alternative hypothesis?At the null hypothesis, it is tested if there is no evidence of different performance, that is, the mean is of 8.5 hours, hence:
[tex]H_0: \mu = 8.5[/tex]
At the alternative hypothesis, it is tested if there is evidence of different performance, with a mean different of 8.5 hours, hence:
[tex]H_1: \mu \neq 8.5[/tex]
What is the test statistic?The population standard deviation is known, hence the z-distribution is used to obtain the test statistic.
The equation is given as follows:
[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.[tex]\sigma[/tex] is the standard deviation of the population.n is the sample size.The parameters for this problem are given as follows:
[tex]\overline{x} = 8.8, \mu = 8.5, \sigma = \sqrt{1.21} = 1.1, n = 21[/tex]
Hence the test statistic is calculated as follows:
[tex]z = \frac{8.8 - 8.5}{\frac{1.1}{\sqrt{21}}}[/tex]
z = (8.8 - 8.5)/(1.11/square root of 21)
z = 1.24.
What is the p-value and conclusion?The p-value is obtained using a z-distribution calculator.
We have a two-tailed test, as we are testing if the mean is different of a value, with a test statistic of z = 1.24, hence the p-value is of:
0.215.
(2 multiplied by the subtraction of one and the p-value of z = 1.24).
The p-value is greater than the significance level of 0.05, meaning that there is not enough evidence to reject the null hypothesis and conclude that there is a difference between the two methods.
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Solve x^2+2x>15 algebraically and graphically, brainliest to the best detailed answer, including the steps.
The algebraic solutions are x < -5 and x > 3.
The graph is given below.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 4 = 9 is an equation.
We have,
x² + 2x > 15
x² + 2x - 15 > 0
x² + (5 - 3)x - 15 > 0
x² + 5x - 3x - 15 > 0
x(x + 5) - 3(x + 5) > 0
(x - 3)(x + 5) > 0
x + 5 > 0 and x - 3 > 0
x < -5 and x > 3
The graph is given below.
Thus,
x < -5 and x > 3 is the solution.
The graph is given below.
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How can you identify and write equivalent expressions?
You can identify equivalent expressions if one of the expressions can be written like the other and it is usually written by combining any like terms on each side of the equation such as x-terms with x-terms and constants with constants.
What is an Equivalent expression?These are the type of expressions that have similar value or worth but do not look the same.
This type of expression can be identified if one of the expressions can be written like the other and example can be seen below:
3(x + 3) and 3x + 9
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if crystals room measures 10.5 ft long and 7 feet wide and 9 feet tall how much paint will be needed to paint her bedroom
Answer:
She needs 18,731.6 Litres of paint to paint her bedroom.
Step-by-step explanation:
Given;Length (l) = 10.5 ftWidth (w) = 7 feetHeight (h) = 9 feetTo Find;How much paint will be needed to paint her bedroom.Formula;V = l × w × hSo,
V = l × w × h
V = 10.5 × 7 × 9 feet
V = 661.5 cubic feet
We know that,
1 cubic foot = 28.317 litresSo,
661.5 × 28.317 = 18,731.6 Litres
Thus, She needs 18,731.6 Litres of paint to paint her bedroom.
PLS HELP I HAVE NO IDEA
Complete the remainder of the table for the given function
y=x/2 + 5
Answer:
Step-by-step explanation:
(-2,4)
(0,5)
(2,6)
(4,7)
Answer: 4, 5, 6, 7,
Step-by-step explanation: From the formula y = x / 2 + 5
substitute x with the value -2
It becomes -2 / 2 + 5
that equals 4
then do the same for the rest
when x = 0
y = 0 / 2 + 5
y = 5
when x = 2
y = 2 / 2 + 5
y = 1 + 5
y = 6
when x = 4
y = 4 / 2 + 5
y = 2 + 5
y = 7
I hope this was helpful
. Marissa is painting one of her bedroom walls. She marks off of the wall. Then she divides the marked section into 3 equal parts and paints one part blue. What fraction of the whole wall is blue?
Answer: Marissa divided the marked section of the wall into 3 equal parts. If she painted one of these parts blue, then the part that she painted blue is 1 out of the 3 equal parts. So, the fraction of the wall that is blue is 1/3.
Another way of understanding this is that if the wall is divided into 3 equal parts and Marissa painted 1 part, then the fraction of the wall that is painted is 1/3.
So the fraction of the whole wall that is blue is 1/3.
Step-by-step explanation:
Let x represent the number of laptops. Let y represent the number of smartphones.
Sully earns $80 by selling a laptop and $60 by selling a smartphone. He wants to make a profit of at least $6,800. Write an inequality to model the amount Sully earns by selling laptops and smartphones
Therefore , As a result, 80x + 60y ≥ 6800 is the answer to the inequality problem given.
What is inequality?Both formulae and inequalities are mathematical phrases created by connecting two expressions. The equal sign (=) in an equation denotes that the two expressions are regarded as equal. In contrast, an inequality indicates that the two phrases are not always equal by using the symbols >,≥,≤, or.<
Here,
Let the quantity of sold laptops, x
y —- the quantity of sold smartphones
The fact that to solve this problem we use inequality help
The problem's underlying inequity is equal to
=> 80x + 60y ≥6800
Therefore , As a result, 80x + 60y ≥ 6800 is the answer to the inequality problem given.
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What is the sum of polynomials 7x 3 4x 2 )+( 2x 3 4x 2?
Therefore , the solution of the given problem of equation comes out to be 9x^3 - 8x^2.
Explain the equation.A mathematical equation is a formula that uses the equal sign (=) to connect two statements and express equality. A mathematical statement that proves the equality of two mathematical expressions is known as an equation in algebra. For instance, the components 3x + 5 and 14 in the equation 3x + 5 = 14 are separated by an equal sign. The equal sign is a crucial part of mathematical formulas, especially equations. Algebra is frequently utilized in equations. Algebra is used in mathematics when it is difficult to determine a precise quantity.
Here,
Polynomial function total
Polynomial functions are those that have a leading degree of three or higher. given the total;
(7x^3-4x^2)+(2x^3-4x^2)
Expand => (7x 3-4 times) + (2x 3-4 times)
= > 7x^3-4x^2 + 2x^3-4x^2
assemble similar terms
=> 7x^3 + 2x^3 - 4x^2 -4x^2
=> 9x^3 - 8x^2
Therefore , the solution of the given problem of equation comes out to be 9x^3 - 8x^2.
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An elliptical mirror measures 16 inches wide and 20 inches high. The ellipse is centered at (0, 52) on a coordinate plane, where units are in inches. Which equation represents the mirror?
(y - 52)²/400 + x²/256 = 1
(y – 52)²/100+ x²/64 = 1
X²/400+ (y-52)²/256 = 1
x²/100+(y-52)²/64 = 1
Equation of ellipse is x²/400 + (y-52) ²/256 =1
What is ellipse?An ellipse is an oval shaped close geometrical figure. Ellipse has two radius, one is horizontal, and the other is vertical. An ellipse can be stretched either horizontal direction or vertical direction.
Which equation represent the mirror?given, width of elliptical mirror = 16 inches
height of elliptical mirror = 20 inches
the center of ellipse = (0, 52)
as the width of elliptical mirror is less than height so the ellipse has vertical major axis.
vertical radius is denoted by b.
and horizontal radius is denoted by a.
as per the question, a = 16 inches and b = 20 inches.
we know the equation of ellipse whose major axis is vertical and center at the origin is shown by
x² / b² + y²/ a² = 1
we are given information about an ellipse having center (0, 52) and vertical radius 20 and horizontal radius 16.
the equation of this ellipse is.
x² / (20) ² + (y -52) ²/ (16)² = 1
x²/400 + (y - 52) ² /256 = 1
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Find the output (y) of the function y= -7+5 if the input is (x) is 15
Y=
Answer: -100
Step-by-step explanation: -7(15)+5
It said x=15 so put 15 in and solve from there and that would be -100
HOPE THIS HELPS ANY I JUST completed FUNCTIONS AND MAN IT WAS FUN
5 1/8+ 6 1/8 in simplest form
Answer: The answer is 11 1/4
Step-by-step explanation:
Willl mark you brainlieestttt
Answer:
√28 and √11
Step-by-step explanation:
Note : -
Irrational number is any real number that cannot be expressed as the quotient of two integers.
Here,
√28 = 5.291.......
5.291.... is an infinite number, it keeps going, it cannot be expressed in terms of a fraction.
Similarly,
√11 = 3.316......
Hence,
√28 and √11 are irrational numbers.
Please help quickly
Nova wants to measure the height of a tree. She asks her brother Caden to stand so that the
tip of his shadow coincides with the tip of the tree’s shadow. She stands at the tip of Caden’s
shadow. Caden, who is 1.2 meters tall, is 4.2 meters from Nova and 6.5 meters from the tree.
The tree is ___ meters high. Round your answer to the nearest tenth and write it as a decimal.
Answer:
The tree is 9.1 meters high.
Step-by-step explanation:
We can use the Pythagorean Theorem to solve this problem. Let T be the height of the tree. We can set up the following equation:
4.2^2 + T^2 = 6.5^2
Solving for T, we get:
T = sqrt(6.5^2 - 4.2^2)
T = sqrt(41.25 - 17.64)
T = sqrt(23.61)
T = 4.878
Since the height of the tree is measured to the nearest tenth, we can round this value to 9.1 meters.
Write and solve an equation to find what fraction of the student f voted for fruit juice or water
If [tex]\frac{1}{4}[/tex] of fruit punch is pineapple juice. About [tex]\frac{1}{6}[/tex] of the fruit punch is orange juice, then fraction of punch that is pineapple juice or orange juice is [tex]\frac{5}{12}[/tex] .
The fraction of pineapple juice in fruit punch is = [tex]\frac{1}{4}[/tex] ;
the fraction of orange juice in fruit punch is = [tex]\frac{1}{6}[/tex] ;
the equation that can be used to find the fraction of fruit punch that is pineapple juice or orange juice is
⇒ [tex]Frcation Of (Pinapple Juice +Orange Juice)[/tex]
substituting the values ,
we get the equation as ;
⇒ [tex]\frac{1}{4} +\frac{1}{6}[/tex]
taking LCM as 4 and 6 as 12 and solving further ,
we get ;
⇒ [tex]\frac{3}{12} +\frac{2}{12}[/tex] ; = [tex]\frac{5}{12}[/tex] .
Therefore , the fraction of fruit punch that is pineapple juice or orange juice is [tex]\frac{5}{12}[/tex] .
The given question is incomplete , the complete question is
About 1/4 of fruit punch is pineapple juice. About 1/6 of the fruit punch is orange juice. Write and solve an equation to find the fraction of the punch that is pineapple juice or orange juice .
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A car is traveling at a speed of 61 kilometers per hour. What is the car's speed in kilometers per minute? How many kilometers will the car travel in 20 minutes? Do not round your answers.
The car's speed in kilometers per minute is 1.01667. And the number of kilometers will the car travel in 20 minutes will be 20.33 kilometers.
What is speed?Speed is defined as the length traveled by a particle or entity in an hour. It is a scale parameter. It is the ratio of length to duration.
We know that the speed formula
Speed = Distance/Time
A car is traveling at a speed of 61 kilometers per hour.
Convert the speed into a kilometer per minute. Then we have
Speed = 61 / 60
Speed = 1.01667 kilometers per minute
Then the number of kilometers will the car travel in 20 minutes will be given as,
1.01667 = d / 20
d = 1.01667 x 20
d = 20.33 kilometers
The car's speed in kilometers per minute is 1.01667. And the number of kilometers will the car travel in 20 minutes will be 20.33 kilometers.
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Which of the following equations describes the nth term for the sequence 3/5,1/5,1/15,1/45,\cdots2
A. an = 1/3 (3/5)n-1
B. an = 1/3 (3/5)n
C. an = 3/5 (1/3)n-1
D. an = 3/5 (1/3)n
The equation which describes the nth term for the sequence 3/5,1/5,1/15,1/45,\cdots2 is an = 3/5 (1/3)n-1 , which is option C.
We use the formula an=a1+(n−1)d an = a1 + (n−1)d , to calculate the nth term of a given sequence. The 'n' here represents a number.
Using the nth term we can any number on the sequence which allows us to locate any term in a series.
By substituting different values for the term number, we can create a series utilizing the nth term (n).
To find the tenth term we use the sequence formula and substitute a random number in place of n.
Example if we need to find the first 50 term then we will substitute n as 50.
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The first laptop ever was sold was the Osborne 1. It came onto the market in 1981, and
it had a mass of 11 kg. Suppose that since 1981 the mass of laptops has decreased at an
average rate of 4. 7% per year. What was the approximate weight of a laptop in 2019?
The Osborne 1 was the first laptop ever released, and it weighed 11 kg in 1981. Since then, the average rate of weight decrease has been 4.7% per year. This means that in 2019, a laptop would have weighed around 2.3 kg.
1981: 11 kg
1982: 10.5 kg (11 kg - 0.55 kg = 10.5 kg)
1983: 10.02 kg (10.5 kg - 0.48 kg = 10.02 kg)
1984: 9.55 kg (10.02 kg - 0.47 kg = 9.55 kg)
2019: 2.3 kg (3.44 kg - 1.14 kg = 2.3 kg)
The Osborne 1 was the first laptop ever released in 1981, and it weighed 11 kg. Since then, the average rate of weight decrease has been 4.7% per year. This means that the mass of a laptop decreases by 0.47 kg every year on average. So, in 1982 it would have weighed 10.5 kg, in 1983 it would have weighed 10.02 kg, and so on. By 2019, the weight of a laptop would have decreased to 2.3 kg as a result of this 4.7% average rate of weight decrease.
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a wheel having a diameter of 50 cm. if the elf travels a distance of 25 meters. how many revolutions did the wheel go through
Approx 16 revolutions the wheel go through to travel the distance 25 meters by a diameter of 50 cm.
In the given question, a wheel having a diameter of 50 cm.
If the elf travels a distance of 25 meters then we have to find how many revolutions did the wheel go through.
Since we have the wheel's diameter, we can determine the radius of the wheel. As we already know, the length of a wheel's revolution is equal to its circumference, which is a circle with a diameter of 2πr.
We will now enter the radius value into the algorithm to determine how many revolutions the wheel must make. To reach the desired result, divide the total distance by the distance covered in one revolution to determine the number of revolutions needed to travel 25 m.
r = [tex]\frac{50}{2}[/tex] = 25 cm
The radius of the wheel is 25 cm.
Now we put the value of r in the formula of circumference.
2πr = 2π×25
2πr = 50π cm
Total distance is 25 m. So we convert m in cm by multiplying the 100.
Now the total distance is 2500 cm.
2500 = n × 50π
Now putting the value of π = 3.14
2500 = n×50 × 3.14
2500 = n×157
Divide by 157 on both side, we get
n = 15.924
n = 16(approx)
Hence, approx 16 revolutions the wheel go through to travel the distance 25 meters by a diameter of 50 cm.
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OK SO, Sonja has a bank account balance of $1400. Each week, her balance changes by -$85.25. She wants to keep the balance above $377.
How many weeks will Sonja's balance remain above $377?
Select from the drop-down menu to correctly complete the statement.
There are 12 weeks will Sonja's balance remain above $377.
What is account balance?
The balance is the sum owed on an account in banking and accounting. The term "balance" in bookkeeping refers to the difference between the total of debit entries and the total of credit entries made into an account over the course of a given financial period. The account shows a debit balance when total debits exceed total credits.
Given:
Sonja has a bank account balance of $1400. Each week, her balance changes by -$85.25. She wants to keep the balance above $377.
We have to find the number of weeks will Sonja's balance remain above $377.
Let, $1400 - $377 = $1023.
⇒ $1023 / $85.25 = 12
Hence, there are 12 weeks will Sonja's balance remain above $377.
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Find the measure of the indicated angle to the nearest degree.
Answer:
70⁰
Step-by-step explanation:
We need to use the knowledge of trigonometry to find the indicated angle. Given the information in relation to the angle we have the opposite and the hypotenuse.
we therefore use Sine in this calculations
Sin ? = opposite/ hypotenuse
= 16/17
= 0.9412
to get the angle we get the sine inverse
Sine inverse of 0.9412 = 70.25⁰
= 70⁰
Answer:
? ≈ 70°
Step-by-step explanation:
using the sine ratio in the right triangle
sin ? = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{16}{17}[/tex] , then
? = [tex]sin^{-1}[/tex] ( [tex]\frac{16}{17}[/tex] ) ≈ 70° ( to the nearest degree )
Use the graph of the function to find the domain and range of f. (Enter your answer using interval notation.)
The domain and range of the functions are Domain: (-∞,∞),{x|x∈R} and Range: [4,∞),{y|y≥4}.
What is vertex form of a quadratic function of graph?The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. The coefficient a determines whether the graph of a quadratic function will open upwards or downwards.
From the graph, vertex is (h, k)=(0, 4) and (x, y)=(2, 0)
Here, y= a(x - h)² + k
0=a(2-0)+4
-4=2a
a=2
Substitute, a=2 and (h, k)=(0, 4) in y= a(x - h)² + k, we get
y=2(x-0)²+4
y=2x²+4
So, f(x)=2x²+4
Find the domain by finding where the function is defined. The range is the set of values that correspond with the domain.
Domain: (-∞,∞),{x|x∈R}
Range: [4,∞),{y|y≥4}
Now,
(a) f(-3)= 2(-3)²+4=18
(b) f(2)= 2(2)²+4=12
(c) f(0)= 2(0)²+4=4
(d) f(3)= 2(3)²+4=18
Therefore, the domain and range of the functions are Domain: (-∞,∞),{x|x∈R} and Range: [4,∞),{y|y≥4}.
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A student has two different samples of soils. She drops a small amount of water on each sample and squeezes the soil in her hand. Her results are shown below. one hand shown holding crumbly soil, the other hand with soil held together and smooth Which statement best describes this test? A. The student is testing each soil's permeability, and soil X is more permeable than soil Y. B. The student is testing each soil's porosity, and soil Y is more porous than soil X. C. The student is testing each soil's texture, and soil Y has more clay than soil X. D. The student is testing each soil's minerals, and soil X has more minerals than soil Y.
The student is testing each soil's permeability, and soil X is more permeable than soil Y. The correct option is A.
What is an experimental approach?An experiment is a technique used to confirm or deny a hypothesis, as well as assess the likelihood or effectiveness of something that has never been tried before. Experiments show what happens when a specific factor is modified, which sheds light on cause-and-effect relationships.
Porosity can be defined as the quality of the material having pores or holes. The permeability can be defined as the state or quality of a material that allows liquids or gases to pass through it.
The soil which does not hold much water can be considered of high porosity and high permeability. As it is only possible if the soil has a large number of pores and it is not in compact form.
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