C(t equation )'s is thus: [tex]C(t) = (t - 0)^2 + 3[/tex] or condensed, [tex]C(t) = t^2 + 3[/tex] as the vertical stretch or compression factor and (h, k) is the vertex.
what is parabola ?A parabola is a symmetry U-shaped plane curve. This graph represents a quadratic function using the formula y = ax2 + bx + c. The parabola's direction and shape are determined by the nonzero integers a and b in this equation, whereas the parabola's location on the x-axis is affected by the linear coefficient b and the constant coefficient c. A parabola has a vertex, or the location where the curve changes direction, and it can open upward or downward.
given
We can start by noting that since the vertex of a parabola always lies in the centre of the two x-intercepts for a quadratic function, the parabola's vertex must be at the coordinates (0, 3).
The equation can be expressed in the vertex form of a quadratic function as follows:
[tex]C(t) = a(t-h)^2 + k[/tex]
where "a" is the vertical stretch or compression factor and (h, k) is the vertex. Using the values from the table as inputs, we obtain:
[tex]C(t) = a(t - 0)^2 + 3[/tex]
We still need to determine what "a" is worth. To do this, we can utilise a different table row, for instance (1, 4):
[tex]4 = a(1 - 0)^2 + 3[/tex]
1 = a
C(t equation )'s is thus: [tex]C(t) = (t - 0)^2 + 3[/tex] or condensed, [tex]C(t) = t^2 + 3[/tex] as the vertical stretch or compression factor and (h, k) is the vertex.
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A certain coin is a circle with diameter 18 mm. What is the exact area of either face of the coin in terms of pie?
Answer:
[tex]81\pi[/tex] [tex]mm^2[/tex]
Step-by-step explanation:
Let's recall the formula for the area of a circle:
[tex]A = \pi r^2[/tex]
We are given the diameter, but the formula uses the radius. Since the radius is equal to one-half of the diameter, we can find the radius by doing this:
[tex]r = \frac{1}{2}d=\\\\r=\frac{1}{2}(18)= \\\\r=9[/tex]
Now that we've found the radius is 9 mm, let's substitute the values into the formula for the area of a circle. We have:
[tex]A = \pi r^2=\\A=\pi (9^2)=\\A=\pi (81)=\\A=81\pi[/tex]
So, we've found that the exact area, in terms of pi, of either face of the coin is [tex]81\pi[/tex] [tex]mm^2[/tex].
To find the area of the coin/a circle use this equation:
(a = area, r = radius, d = diameter)
[tex]\text{a = r}^2[/tex]
So we need to do for the radius.
[tex]\text{r} = \dfrac{\text{d}}{2}[/tex]
[tex]\text{r} = \dfrac{18}{2}[/tex]
[tex]\text{r} = 9[/tex]
Then solve
[tex]\text{a = 9}^2[/tex]
[tex]\boxed{\bold{a = 81}}[/tex]
Create a Truth Table for
(A ⋀ B) → C
The truth table is given above for (A ⋀ B) → C.
What is the logical statement?
A logical statement, also known as a proposition or a statement of fact, is a declarative sentence that is either true or false, but not both. It is a statement that can be evaluated based on the available information or evidence to determine its truth value. In other words, a logical statement is a statement that can be either true or false, but not both.
To create a truth table for the logical statement (A ⋀ B) → C, we need to consider all possible combinations of truth values for propositions A, B, and C.
There are 2 possible truth values (true or false) for each proposition, so there are 2³ = 8 possible combinations.
We can organize these combinations into a table as follows:
| A | B | C | (A ⋀ B) | (A ⋀ B) → C |
|---|---|---|---------|-------------|
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | F | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | T | F | F | T |
| F | F | T | F | T |
| F | F | F | F | T |
In this table, the column labeled (A ⋀ B) represents the truth value of the conjunction of A and B (i.e., A AND B), and the column labeled (A ⋀ B) → C represents the truth value of the conditional statement (A ⋀ B) → C.
The symbol "T" represents "true" and the symbol "F" represents "false".
Hence, The truth table is given above for (A ⋀ B) → C.
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2. Violet is baking cupcakes for a bakesale. The equation for her profit, p, based on the
number of cupcakes she sells, c, is based on the equation p = 2.75c-24. What is the best
nterpretation of the number -24.
A. How many cupcakes she sold.
B. How much it cost to buy the ingredients for the cupcakes.
C. How much each cupcake cost.
D. What her profit is.
Answer: The best interpretation of the number -24 in the equation p = 2.75c-24 is option D: What her profit is.
The equation is in slope-intercept form, where the coefficient of c (2.75) represents the profit per cupcake and the constant term (-24) represents the fixed costs or expenses that Violet incurs regardless of how many cupcakes she sells.
In this case, the constant term of -24 represents the fixed costs such as the cost of ingredients, supplies, and other expenses that Violet incurs to make the cupcakes. This cost is subtracted from the total revenue generated by selling cupcakes to determine the profit. Therefore, the number -24 represents the fixed costs or expenses and its inclusion in the equation allows us to determine Violet's profit as a function of the number of cupcakes sold.
Step-by-step explanation:
Can you find X? Show how did u find
Answer:
x=90°
Step-by-step explanation:
As in the angle, there is a box giving us info that x has to be 90°.
But to be sure that there is no mistake we have to do the following:
Look at all the other angles (see what kind of angles they are).Add all the angles up to 360°( in this case as the angle we are looking for is on a straight line which gives straight line=180°).Checking and comparing the two answers.So we are looking at the surroundings of angle x (which is on a straight line) we see that it is a right angle and look at the angle on the same line is a right angle too.
The equation right angle=90° helps us see that because there are two right angles on a 180° line (90°+90°+180°).
Therefore the answer is:
x=90°
1
2 3
5
1
Jared takes 10 minutes to wash dishes and 20
minutes to write a paper. Jason takes 10 minutes to
wash dishes and 30 minutes to write a paper. Which
of the following statements is correct?
Summary
O Jared has a comparative advantage in washing
dishes.
O Jared has absolute advantage in writing the
paper.
Back
O Jared has absolute advantage in washing the
dishes.
O Jared has a comparative advantage in writing a
paper.
Next
The correct answer is Jared has absolute advantage in washing the dishes.
I need help with this please
3. Choose the best answer.
The ABC Ad Agency has two automotive tire chains as clients. This could lead to a(n)
A)conflict of interest
B)defect
C)stereotype
D)overworked staff
The correct option - A) conflict of interest is caused due to ABC Ad Agency has two automotive tire chains as clients.
Explain about the conflict of interest:A conflict of interest arises when a person's personal interests, such as those related to their family, friends, finances, or social standing, potentially impair their judgement, choices, or actions at work. Conflicts of interest are taken seriously enough by government organisations that they are governed.
Conflicts of interest happen most frequently when requirements and interests collide. Because of the nature of connections against policies of organisations or regulations of the federal and state levels, various forms of conflicts of interest may arise. Humans are prone to prejudice (having an unfair preference) due to trivial considerations like friendship, food, or flattery. They can also be persuaded to make a choice by the prospect of gaining money, status, or power.For the given data- The ABC Ad Agency has two automotive tire chains as clients.
This, could lead to a(n) conflict of interest.
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Graph X-3<8 and 6x<72
Answer:
x < 11 & x < 12
Step-by-step explanation:
x - 3 < 8 and 6x < 72
x < 11 & x < 12
How can I solve this math problem within 5 minutes
The equation a² + b² = c² represents the Pythagorean theorem.
What is the Pythagorean theorem?This is a theorem that states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
The Pythagorean theorem is used to find the length of one of the sides of a right-angled triangle when the lengths of the other two sides are known.
To solve the problem, you need to have at least two side lengths given. Then, you can use the equation to find the length of the third side. For example:
If you know the lengths of sides a and b and need to find the length of the hypotenuse (c):
Plug in the known values for a and b into the equation and solve for c.
Example: If a = 3 and b = 4, then 3² + 4² = c², which gives 9 + 16 = c², so c² = 25, and c = sqrt(25) = 5.
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In the year 2000, the population of Mexico was about 100 million, and it was growing by 1.53% per year. At this growth rate, the function f(x) = 100(1.0153)x gives the population, in millions, x years after 2000. Using this model, in what year would the population reach 106 million? Round your answer to the nearest year.
Answer:
2004
Step-by-step explanation:
A set of exam scores is normally distributed and has a mean of 74.4 and a standard deviation of 8.3. What is the probability that a randomly selected score will be between 63 and 66?
The probability that a randomly selected score will be between 63 and 66 is approximately 0.0718, or 7.18%.
What is mean?
In statistics, the mean is a measure of central tendency, which is a way of describing the typical or central value of a set of data. The mean is also known as the average, and it is calculated by adding up all the values in a set of data and then dividing by the number of values in the set.
To find the probability that a randomly selected score will be between 63 and 66, we need to calculate the z-scores for these values and then find the area under the normal curve between these z-scores.
The z-score for a score of 63 is:
z = (63 - 74.4) / 8.3
z = -1.37
The z-score for a score of 66 is:
z = (66 - 74.4) / 8.3
z = -1.01
We can use a standard normal distribution table or calculator to find the area under the normal curve between these z-scores.
Using a standard normal distribution table, we find that the area to the left of a z-score of -1.01 is 0.1562, and the area to the left of a z-score of -1.37 is 0.0844. To find the area between these z-scores, we subtract the area to the left of -1.37 from the area to the left of -1.01:
P(-1.37 < z < -1.01) = 0.1562 - 0.0844 = 0.0718
So the probability that a randomly selected score will be between 63 and 66 is approximately 0.0718, or 7.18%.
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Find the number that makes the ratio equivalent to 36:84?
Answer: 3:7
Step-by-step explanation: since the simplest form of the fraction 36/84 is 3/7 that means 36:84 in simplest form is 3:7.
Which of the following is an even function?
f(x) = (x - 1)^2
f(x) = 8x
f(x) = x^2-x
f(x) = 7
Answer: An even function is a function that satisfies the condition:
f(-x) = f(x)
Let's check which of the given functions satisfies this condition:
f(x) = (x - 1)^2
f(-x) = (-x - 1)^2 = x^2 + 2x + 1
f(x) = (x - 1)^2
The two expressions are not equal, so f(x) is not an even function.
f(x) = 8x
f(-x) = -8x = -f(x)
f(x) = 8x
The two expressions are equal with opposite signs, so f(x) is an odd function.
f(x) = x^2 - x
f(-x) = (-x)^2 - (-x) = x^2 + x
f(x) = x^2 - x
The two expressions are not equal, so f(x) is not an even function.
f(x) = 7
f(-x) = 7 = f(x)
f(x) = 7
The two expressions are equal, so f(x) is an even function.
Therefore, the only even function among the given functions is:
f(x) = 7.
Step-by-step explanation:
Samuel went to the grocery store and purchased cans of soup and frozen dinners. Each can of soup has 300 mg of sodium and each frozen dinner has 450 mg of sodium. Samuel purchased 5 more frozen dinners than cans of soup and they all collectively contain 6000 mg of sodium. Write a system of equations that could be used to determine the number of cans of soup purchased and the number of frozen dinners purchased. Define the variables that you use to write the system.
the system of equations is: [tex]300x + 450y = 6000[/tex]
[tex]y = x + 5[/tex]
What best define about variables?Let's define the variables:
Let "x" be the number of cans of soup purchased by Samuel.
Let "y" be the number of frozen dinners purchased by Samuel.
According to the given information, each can of soup has 300 mg of sodium and each frozen dinner has 450 mg of sodium. Samuel purchased 5 more frozen dinners than cans of soup, so we can write the following equations:
The total sodium from cans of soup: 300x mg
The total sodium from frozen dinners: 450y mg
The total sodium from both cans of soup and frozen dinners: [tex]6000 mg[/tex]
So the first equation is:
[tex]300x + 450y = 6000[/tex] (since the total sodium from cans of soup and frozen dinners is 6000 mg)
Since Samuel purchased 5 more frozen dinners than cans of soup, we can write the second equation:
[tex]y = x + 5[/tex] (since y is 5 more than x)
Therefore, the system of equations is:
[tex]300x + 450y = 6000[/tex]
[tex]y = x + 5[/tex]
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Problem 1: Find the Area and round to the nearest tenth.
Answer:
39.96
Step-by-step explanation:
the shape is a parallelogram ao the formula is base x height
A=10.8 x 3.7
A=39.97
Suppose that the functions fand g are defined as follows.
f(x)=2x-1
g(x)=√3x-5
a. i. The function f(x)/g(x) = (2x - 1)/√(3x - 5)
ii. The domain is x > 5/3
b. i. The function f(x) - g(x) = (2x - 1) - √(3x - 5)
ii. The domain is x > 5/3
What is a function?A function is a mathematical relation ship between two variables.
Since we have the functions f and g defined as follows
f(x) = 2x-1
g(x) = √3x-5
a. i To find f/g we note that
(f/g)(x) = f(x)/g(x)
So, substituting the values of the variables into the equation, we have that
f(x)/g(x) = (2x - 1)/√(3x - 5)
ii. The domain of f(x)/g(x) = (2x - 1)/√(3x - 5) is the value for which the denominator g(x) > 0.
So,g(x) > 0
⇒ √(3x - 5) > 0
⇒ 3x - 5 > 0²
⇒ 3x > 5
⇒ x > 5/3
So, the domain is x > 5/3
b. i. to find f - g, we note that
f - g = f(x) - g(x)
So, substituting the values of the variables into the equation, we have that
f(x) - g(x) = (2x - 1) - √(3x - 5)
ii. The domain of f(x) - g(x) is the value of x at which g(x) > 0
So. g(x) > 0
⇒ √(3x - 5) > 0
⇒ [√(3x - 5)]² > 0
⇒ 3x - 5 > 0
⇒ 3x > 5
⇒ x > 5/3
So, the domain is x > 5/3
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lisa ran 1/2 of a mile.jan ran 3/6 of a mile.which girl ran further
The fraction that has been given illustrates that the person who ran further is Lisa and Jane.
How to solve fractionYour information isn't complete. Therefore, an overview of the fraction will given.
Let's assume that Lisa ran 1/2 of a mile and Jane ran 3/6 of a mile. In order to know who ran more, you can convert the fraction to percentage.
This will be
[tex]\text{Lisa} = \dfrac{1}{2} \times 100 = \bold{50\%}[/tex]
[tex]\text{Jane} = \dfrac{3}{6} \times 100 = \bold{50\%}[/tex]
Therefore, both Lisa and Jane ran more.
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in a kite, the measurements of the angles are 3x, 75, 90, and 120. Find the value of x. What are the measures of the angles that are congruent
x is 25 and the congruent angles in the kite are 25 and 155 degrees.
What is quadratic equation?
it's a second-degree quadratic equation which is an algebraic equation in x.
In a kite, the two pairs of adjacent angles are congruent. Let's call the congruent angles x and y:
x y
+---+ +---+
/ \ / \
+-------+-------+
\ / \ /
+---+ +---+
y x
We know that the measurements of the angles in the kite are 3x, 75, 90, and 120, so we can set up an equation based on the sum of the angles in a quadrilateral:
3x + 75 + 90 + 120 = 360
Simplifying this equation, we get:
3x + 285 = 360
Subtracting 285 from both sides, we get:
3x = 75
Dividing both sides by 3, we get:
x = 25
So x is 25. To find y, we know that the sum of x and y must be 180 (because they are congruent angles that add up to a straight line). So we can set up another equation:
x + y = 180
Substituting x = 25, we get:
25 + y = 180
Subtracting 25 from both sides, we get:
y = 155
Therefore, x is 25 and the congruent angles in the kite are 25 and 155 degrees.
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Zoey is buying 6 pairs of work gloves.
She has a coupon for $2 off the regular
price of each pair. After using the
coupon, the total cost was $47.94.
Which equation can be used to find the
original cost of a pair of gloves?
The original cost of a pair of gloves is 3.9
What is the meaning of [tex]s_{i-j}[/tex]?
This expression [tex]s_{i-j}[/tex] describes the distance between two points i and j in a geometric object
Explaining the meaning of the expressionIn the context of symmetry and rotations, [tex]s_{i-j}[/tex] typically refers to the distance between two points i and j in a geometric object, such as a crystal lattice or a molecule.
It is a vector that points from point i to point j, and its magnitude is the distance between the two points.
The distance vector [tex]s_{i-j}[/tex] is also used to describe the position of a point in the crystal lattice relative to the rotation axis.
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A skydiver jumps out of a plane from a certain height. The graph below shows their height h in meters after t seconds. How long is the skydiver in the air? Height (in meters) 3000 2500 2000 h 1500 1000 500 0 (0, 2592.1) 2 4 6 8 10 12 14 16 Time (in seconds) 18 20 (23, 0) t 22 24
The skydiver is in the air for 23 seconds.
What is graph?A diagram or pictorial representation that organises the depiction of facts or values is known as a graph.
The relationships between two or more items are frequently represented by the points on a graph.
The skydiver is in the air as long as their height is greater than zero. From the graph, we can see that the skydiver reaches a height of zero at t = 23 seconds. Therefore, the skydiver is in the air for:
t = 23 seconds - 0 seconds = 23 seconds
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2À candy company claims that its jelly bean mix contains 15% blue jelly beans. Suppose that the candies are packaged at random in small bags containing about 200 jelly beans. What is the probability that a bag will contain more than 20% blue jelly beans?
We find that P(Z > 2.46) is roughly 0.007 using a calculator or a basic normal distribution table. The probability that a bag will contain more than 20% blue jellybeans is therefore approximately 0.007 or 0.7%.
Define probability?The probability of an event is the ratio of good outcomes to all other potential outcomes. The number of successful outcomes for an experiment with 'n' outcomes can be expressed using the symbol x.
Here in the question,
We can utilise the binomial distribution formula to resolve this issue. In a bag of 200 jelly beans, let X represent the proportion of blue jelly beans. Following that, X exhibits a binomial distribution with parameters of n = 200 and p = 0.15, where p is the likelihood of drawing a blue jellybean.
The formula for determining the likelihood of finding more than 20% blue jellybeans in a bag is:
P (X > 0.2 × 200) = P (X > 40)
Since n is large (200) and p is not too near to 0 or 1, we can utilise the usual approximation to the binomial distribution. We may determine the equivalent mean and standard deviation of the normal distribution by using the mean and variance of the binomial distribution:
μ = np = 200 × 0.15 = 30
σ = √ (np(1-p)) = √ (200 × 0.15 × (1-0.15)) = 4.07
Then, we can standardize the random variable X as:
Z = (X - μ) / σ
So, we have:
P(X > 40) = P((X - μ) / σ > (40 - μ) / σ)
= P(Z > (40 - 30) / 4.07)
= P(Z > 2.46)
We find that P(Z > 2.46) is roughly 0.007 using a calculator or a basic normal distribution table. The likelihood that a bag will contain more than 20% blue jellybeans is therefore approximately 0.007 or 0.7%.
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In Math town,60% of the population are males and 30% of them have brown eyes. Of the total math town population 28 % have brown eyes. What percentage of the females in math town have brown eyes?
A) 20%
B) 24%
C) 25%
D) 28%
thought i would leave this but ⇒ty to whoever answers this, have an amazing day <3
The percentage of females with brown eyes as follows is 25%.
What is percentage?Percentage is a way of expressing a number as a fraction of 100. It is represented by the symbol "%". The word "percent" means "per hundred".
According to question:Let's assume that the total population of Math town is 100 people for the sake of simplicity. Then, we can use the following information to create a table:
Population Males Females
Total 60 40
Brown eyes 18 ?
From the table, we can see that 60% of the population are males, so there are 60 x 0.3 = 18 males with brown eyes.
We also know that the total population with brown eyes is 28%, so there are 100 x 0.28 = 28 people with brown eyes. Therefore, there must be 28 - 18 = 10 females with brown eyes.
Finally, we can calculate the percentage of females with brown eyes as follows:
% of females with brown eyes = (10 / 40) x 100% = 25%
Therefore, the answer is (C) 25%.
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12×(3+2²)÷2-10 what is the answer
Answer:
32 I hope this helps please make me a brianlist that would help :)
Please Explain: Jana determined that she is approximately 1.013 x 10^-3 miles tall. How many places, and in which direction, must you move the decimal to determine her height in standard notation?
The states C and D spend a total of $56.8 million for tourism. The state C spends $1.8 million more than the state D. Find the amount that each state spends on tourism.
D spends $___ million on tourism , and state C spends $___million on tourism.
Let's denote the amount that state D spends on tourism as "x" million dollars. According to the given information, state C spends $1.8 million more than state D, so state C would spend "x + 1.8" million dollars on tourism.
The total amount spent by state C and state D combined is $56.8 million. So we can write the equation:
x + (x + 1.8) = 56.8
Now, we can solve this equation to find the values of x and x + 1.8, which represent the amount spent by state D and state C, respectively.
x + x + 1.8 = 56.8
2x + 1.8 = 56.8
2x = 56.8 - 1.8
2x = 55
x = 55 / 2
x = 27.5
So, state D spends $27.5 million on tourism, and state C spends $27.5 + $1.8 = $29.3 million on tourism.
Jasper's aunt gave him a big bin of 500 beads made out of assorted materials to use for the wind chimes he makes. Jasper takes out a handful of beads, looks at the types of beads, then puts them back. Here are the materials of the handful he selected: glass, clay, wood, glass, wood, clay, metal, clay, wood, glass, wood, clay, metal, wood, clay Based on the data, estimate how many glass beads are in the bin. If necessary, round your answer to the nearest whole number.
We can estimate that there are approximately 134 glass beads in the bin.
What is probability?
Probability is a measure of the likelihood of an event occurring.
To estimate the number of glass beads in the bin, we can use the proportion of glass beads in the handful that Jasper selected.
There are 15 beads in the handful, and 4 of them are glass. So, the proportion of glass beads in the handful is:
4/15 ≈ 0.267
We can assume that the proportion of glass beads in the bin is similar to the proportion in the handful. Therefore, we can estimate the number of glass beads in the bin as
0.267 x 500 ≈ 134
Therefore, we can estimate that there are approximately 134 glass beads in the bin.
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What is the value of x in this equation?
2x + 6
5
= −1(x - 11)
Answer:
To solve the equation 2x + 6/5 = -1(x - 11), we can use the distributive property of multiplication over addition/subtraction to simplify the right-hand side of the equation:
2x + 6/5 = -x + 11
Next, we can rearrange the terms so that all the x terms are on one side of he equation and all the constant terms are on the other side:
2x + x = 11 - 6/5
Combining like terms and simplifying the right-hand side, we get:
3x = 49/5
Dividing both sides by 3, we get:
x = 49/15
Therefore, the value of x in the equation 2x + 6/5 = -1(x - 11) is 49/15.
x f(x) f ′(x) g(x) g′(x)
2 5 1 2 1
4 −2 −6 5 4
5 1 2 4 5
6 5 1 2 −2
The functions f and g have continuous derivatives. The table gives values of f, f ′, g, and g′ at selected values of x.
Part A: Find h′(2) if h(x) = g(f(x)). (5 points)
Part B: Find m′(2) if m(x) = f(x2). (5 points)
Part C: Let k(x) = f(g(x)). Write an equation for the line tangent to the graph of k at x = 4. (10 points)
Part D: Let j of x is equal to g of x divided by f of x. Find j′(2). (10 points)
Answer:
A. h'(2) = 5
B. m'(2) = -24
C. y = 8x - 31
D. j'(2) = 3/25 = 0.12
Step-by-step explanation:
When differentiating composite functions, use the chain rule.
[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule for Differentiation}\\\\$\left[f\left(g(x)\right)\right]'=f'\left(g(x)\right) \cdot g'(x)$\\\end{minipage}}[/tex]
Chain Rule: The derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.
Part AUsing the chain rule, if h(x) = g(f(x)) then h'(x) = g'(f(x)) ⋅ f'(x).
To find h'(2), substitute x = 2 into the differentiated equation:
[tex]\begin{aligned}h'(x)& = g'(f(x)) \cdot f'(x)\\\\\implies h'(2)& = g'(f(2)) \cdot f'(2)\\& = g'(5) \cdot 1\\& = 5 \cdot 1\\&=5\end{aligned}[/tex]
Therefore, h'(2) = 5.
Part BUsing the chain rule, if m(x) = f(x²) then m'(x) = f'(x²) ⋅ 2x.
To find m'(2), substitute x = 2 into the differentiated equation:
[tex]\begin{aligned}m'(x) &= f'(x^2) \cdot 2x\\\\\implies m'(2) &= f'(2^2) \cdot 2(2)\\&=f'(4) \cdot 4\\&=-6 \cdot 4\\&=-24\end{aligned}[/tex]
Therefore, m'(2) = -24.
Part CTo find the slope of the tangent line to the graph of k at x = 4, substitute x = 4 into the derivative of k(x).
If k(x) = f(g(x)) then k'(x) = f'(g(x)) ⋅ g'(x).
Therefore, the slope of the tangent line is:
[tex]\begin{aligned}k'(x) &= f'(g(x)) \cdot g'(x)\\\\\implies k'(4) &= f'(g(4)) \cdot g'(4)\\&= f'(5) \cdot 4\\&= 2 \cdot 4\\&= 8\end{aligned}[/tex]
Now calculate k(x) when x = 4:
[tex]\begin{aligned}k(x)&=f(g(x))\\\\\implies k(4) &= f(g(4))\\&=f(5)\\&=1\end{aligned}[/tex]
To write the equation of the tangent line, substitute the found slope m = 8 and point (4, 1) into the point-slope equation:
[tex]\begin{aligned}y-y_1&=m(x-x_1)\\y-1&=8(x-4)\\y-1&=8x-32\\y&=8x-31\end{aligned}[/tex]
Therefore, an equation for the line tangent to the graph of k at x = 4 is:
y = 8x - 31Part DTo find the derivative of j(x) use the quotient rule.
[tex]\textsf{If\;\;$j(x)=\dfrac{g(x)}{f(x)}$\;\;then:}\\\\\\j'(x)=\dfrac{f(x)g'(x)-g(x)f'(x)}{(f(x))^2}[/tex]
To calculate j'(2), substitute x = 2 into the equation:
[tex]\begin{aligned} \implies j'(2)&=\dfrac{f(2)g'(2)-g(2)f'(2)}{(f(2))^2}\\\\&=\dfrac{5 \cdot 1-2 \cdot 1}{(5)^2}\\\\&=\dfrac{5-2}{25}\\\\&=\dfrac{3}{25}\\\\&=0.12\end{aligned}[/tex]
Therefore, j'(2) = 3/25 = 0.12.
A truck is delivering goods to a supermarket in Sharjah.
The shopkeeper has run out of cornflakes and he
needs 37 large cartons. One carton contains 22 boxes
of cornflakes. Find at average dimensions of one large
box of cornflakes.
(a) What will be the dimensions of one carton of
cornflakes presuming they are packed in so there
no empty space.
(b) What volume would the truck need to be to deliver
the 37 cartons to the shop in one delivery?
Inorder to find the average dimensions of one large box of cornflakes, we need more information such as the total volume or weight of a large carton of cornflakes. Without that information, we cannot calculate the average dimensions.
What is average means?Average refers to the central value of a set of numbers. It is calculated by adding up all the values in the set and then dividing by the total number of values. The average is also known as the mean.
Define volume?Volume refers to the amount of space an object occupies in three dimensions, typically measured in cubic units such as cubic meters or cubic feet. It is a physical quantity that describes the size of an object or space.
(a) Assuming the boxes are packed tightly with no empty space, the dimensions of one carton of cornflakes can be calculated by multiplying the dimensions of one box by the number of boxes in a carton. However, without knowing the dimensions of one box or the total volume of a carton, we cannot calculate this.
(b) To calculate the volume required for the truck to deliver 37 cartons of cornflakes, we need to know the volume of one carton and multiply it by the number of cartons. Without that information, we cannot calculate the required volume for the truck.
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