Answer:
its C
Step-by-step explanation:
it cant be b because there would be no longest side
neither d or a because ive done the math for it
if f(x)=3^x+10x and g(x)=2x-4 find (f-g) (x) (urgent)
Answer:
(f - g)(x) = [tex]3^{x}[/tex] + 8x + 4
Step-by-step explanation:
(f - g)(x)
= f(x) - g(x)
= [tex]3^{x}[/tex] + 10x - (2x - 4) ← distribute parenthesis by - 1
= [tex]3^{x}[/tex] + 10x - 2x + 4 ← collect like terms
= [tex]3^{x}[/tex] + 8x + 4
the amount of bacteria in a fish tank was 830. After 5 hours, the amount of bacteria is 3202.
Complete the table to determine which of these equations could approximate the amount of bacteria
Answer: 37.92
Step-by-step explanation: We know that 3202 / 5 = 640.4.
We subtract that by 830 to get (830 - 640.4 =) 189.6. Then we do / by five again to get 37.92. That is the answer.
If triangle abc is reflected over the x-axis, reflected over the y-axis, and rotated 180 degrees, where will point b' lie?
Answer:
Step-by-step explanation:
The triangle would end up back where it started. It is hard to explain without a graph. If you have graph paper, you might want to try drawing this out. Say that the original points are at A (1,2) B (2,0) and C(0,0). Now, when we reflect the points over the x axis, they will be the same distance below the x that the points original were about the x axis. Since A was 2 units above the axis, it will now be 2 units below at (1, -2). Points B and C will stay on the x axis and will remain in place at B(2.0) and C(0,0). Since these points are on the line, they were not above the x axis, so they will now not be below the x axis.
Now, we are going to reflect the triangle over the y axis. Since C (0,0) is already on the y axis, it will not move. It will remain there. Since B(2,1) is two units to the right of the y axis, when we flip it, it will now be 2 units to the left of the y axis B (-2,0). Point C will move one unit to the left of the y axis to become (-1,2).
The last thing left it to rotate this final triangle 180 degrees. Since a circle is 360 degrees and 180 is half of a circle, it does not matter if we rotate clockwise or counter-clockwise. If you could trace our new triangle and put a plus sign at the origin (0,0). You would put your pencil on the origin and rotate the two turns at the plus sign. This would put your triangle right back to the beginning. So the original value of B would be the same. In this case C ((2,0)
The point B' would be at point B. The position of point won't change.
What is a reflection?A reflection is a type of transformation that involves flipping a shape, known as the preimage, over a line, known as the line of reflection, to produce a new shape (called the image). You can picture what would happen if you flipped the form over the line in order to graph a reflection.
Given:
Triangle ABC is reflected over the x‐axis, reflected over the y‐axis, and rotated 180 degrees.
Let (x, y) be the coordinates of B.
To find B':
If a figure reflected over x-axis, then
(x, y) → (x, -y)
So, let point B₁ is (x, -y).
If a figure reflected over y-axis, then
(x, y) → (-x, y)
So, (x, -y) → (-x, -y)
So, let point B₂ is (-x, -y).
If a figure rotated 180 degrees about the origin, then
(x, y) → (-x, -y)
(-x, -y) → (x, y)
So the B' is (x, y).
Therefore, the position of point won't change.
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A teacher with 10 students has 30 lesson times available. She will teach exactly one of her students during each lesson time. How many ways are there for her to decide which student she will teach during each lesson time if she must teach each student exactly 3 times
There are [tex]\frac{30!}{(3!)^{10} }[/tex] ways for the teacher to decide which student she will teach during each lesson time if she must teach each student exactly 3 times. Here, "!" represents the factorial.
A number's factorial is the result of multiplying the integer by each natural number below it. Factorial can be symbolized by the letter "!". Thus, n factorial is denoted by n! and is the result of the first n natural numbers.
A whole number's "n" factororial is the sum of that number and each whole number up to one.
When a question asks you to determine how many different ways you can arrange or order a given number of items, you use a factorial.
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Determine which of these sets are subsets of which other of these sets. Check ALL correct answers below.
The Set C is the subset of A and D.
According to the statement
we have given that the three sets which are A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}.
And we have to find that the subsets of these sets.
So,
For a set S to be a subset of a set T, all of the elements of the set S need to be contained in the set T. Firstly, all of the sets are their own subsets, that is, A⊆A,B⊆B,C⊆C,D⊆D.
It follows that B⊂A because of 2∈A and 6∈A. Since 2 is not belongs to C and D. that's why B is not a subset of C and B is not a subset of D.
Taking into account that 4∈A and 6∈A, we conclude that C⊂A. By analogy, since 4∈D and 6∈D, we conclude that C⊂D.
Since |A|>|B|∣A∣>∣B∣ and |A|>|C|,∣A∣>∣C∣, we conclude that A is not a subset of B and A is not a subset of C.
Taking into account that |D|>|B|∣D∣>∣B∣ and |D|>|C|,∣D∣>∣C∣, we conclude that D is not a subset of B and D is not a subset of C. Since,2 not belongs to D that's why A is not a subset of D.
Since 8 not belongs to A, D is not a subset of A.
Overall, The Set C is the subset of A and D.
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Disclaimer: This question was incomplete. Please find the full content below.
Question:
Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets.
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Given f(x) = x² + 3x -3 and g(z) = 2x, find f(g(x)).
Solution
First, we need to decide which function will go inside which. Since
our function reads "f of g of z", we know that we are going to
substitute the function for g(x) into every a spot in the f(x)
function.
f(g(x))=(x
)²+3(x
)-3
Next, perform normal order of operations for each term. (Hint:
(2x)² = (2x)(2x))
f(g(x)) = 4
²+3
2-3
Let's look at how this example would be different if it was
performed in the opposite order. Given f(x) = x² + 3x - 3 and
g(x)=2x, find g(f(x)).
In this case, we'll need to plug the whole function for f(x) into the
input (2) spot in the g(x) function.
g(f(x)) = 4
(x² + 3x - 3)
Then we'll distribute to each term.
g(f(x)) = 4(x²)+2(3x)-3 3
Simplify to get the final answer.
The composition of the functions f(x) and g(x) gives the equation:
[tex]f(g(x)) = 4x^2 + 6x - 3[/tex]
How to get the composition of functions?Here we want to get the composition of the two functions:
[tex]f(x) = x^2 + 3x - 3\\\\g(x) = 2x[/tex]
f(x) is a quadratic and g(x) is a linear equation.
Now we want to get the composition:
[tex]f(g(x))[/tex]
This means that we need to evaluate function f(x) in g(x), so we can replace all the "x" in the function f(x) by the notation "g(x)"
[tex]f(g(x)) = g(x)^2 + 3*g(x) - 3[/tex]
Now we replace all the "g(x)" by the actual function g(x) = 2x, we wll get:
[tex]f(g(x)) = g(x)^2 + 3*g(x) - 3 = (2x)^2 + 3*(2x) - 3[/tex]
finally, we can simplify this to get the composition, which is a quadratic function just like f(x).
[tex]f(g(x)) = 4x^2 + 6x - 3[/tex]
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d) y = 3x - 9 7) 2) y-int. m = x-int. 3) -
**Disclaimer** Hi there! I assumed the question to be finding the "y-intercept", "x-intercept", and "slope/gradient" of the equation "y = 3x - 9". The following answering will be according to this understanding. If it is incorrect, please tell me and I will modify my answer.
Answer:
y-intercept = (0, -9)
x-intercept = (3, 0)
slope (m) = 3
Step-by-step explanation:
What is being asked: to find the corresponding values
y-interceptx-interceptSlope/gradientGiven equation
y = 3x - 9
PART I: y-interceptConcept
The y-intercept of a line is on the coordinate of (0, a), where [ a ] is a constant.
Substitute values into the equation
y = 3x - 9
y = 3 (0) - 9
y = 0 - 9
y = -9
Therefore, the y-intercept is [tex]\Large\boxed{(0, -9)}[/tex]
PART II: x-interceptConcept
The x-intercept of a line is on the coordinate of (a, 0), where [ a ] is a constant.
Substitute values into the equation
y = 3x - 9
0 = 3x - 9
Add 9 on both sides
0 + 9 = 3x - 9 + 9
9 = 3x
Divide 3 on both sides
9 / 3 = 3x / 3
x = 3
Therefore, the x-intercept is [tex]\Large\boxed{(3,0)}[/tex]
PART III: Slope/GradientConcept
In the linear formula, y = mx + b, [ m ] is the slope and [ b ] corresponds to the y-intercept.
Determine the slope
Given the equation, y = 3x - 9, when corresponds to the general linear equation formula:
y = mx + b
y = 3x - 9
Therefore, the slope is [tex]\Large\boxed{m=3}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Emanuel used the calculations below to find the product of the given fractions. (Three-fifths) (StartFraction 4 over 9 EndFraction) (Negative one-half)
The correct option is Step 1 StartFraction (3) (4) (negative 1) over (5) (9) (negative 2) EndFraction
Emanuel did a mistake in her first step by taking negative sign twice.
What is multiplicative rule with different sign ?Positive results are obtained if the sign are same. If the signs disagree, the outcome is adverse. Addition: Keep in mind that a signed number's magnitude and absolute value are the same.
Multiplication and division appear to be more difficult than addition and subtraction, but they are actually far less challenging. The result of multiplying two positive or two negative numbers with the same sign, according to the rule, will always be positive.
For instance:
8 x 4 = 32(-8) x (-4) = 3210 x 9 = 90(-10) x (-9) = 90According to question,
= [tex]\left(\frac{3}{5}\right)\left(\frac{4}{9}\right)\left(-\frac{1}{2}\right)[/tex]
Emanuel found the solution, but she erred in the first step. She incorrectly distributes the negative sign with both 1 and 2, as she should. We can write negative sign with either 1 or 2 but not both.
Correct steps are:
Step 1:
[tex]\frac{(3)(4)(-1)}{(5)(9)(2)}[/tex]
Step 2:
[tex]\frac{-12}{90}[/tex]
Step 3:
[tex]-\frac{2}{15}[/tex]
Therefore, the step 1 was miscalculated by Emanuel.
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The complete question is -
Emanuel used the calculations below to find the product of the given fractions. (Three-fifths) (StartFraction 4 over 9 EndFraction) (Negative one-half)
Step 1 StartFraction (3) (4) (negative 1) over (5) (9) (negative 2) EndFraction Step 2 StartFraction negative 12 over negative 90 EndFraction
Step 3 StartFraction 12 over 90 EndFraction
Step 4 StartFraction 2 over 15 EndFraction
In which step did his first error occur?
Instructions: Find the lengths of the other two sides of the isosceles triangle
Answer:
x=5
h = 5 sqrt 2
Step-by-step explanation:
isosceles right triangles are always 45-45-90, ratio of those triangles are always x:x:x*sqrt 2
what is the answer to this question out of the options given
Answer:
(c) (28/3)√6
Step-by-step explanation:
Side ratios of "special triangles" are used to solve this problem.
30-60-90° triangle — 1 : √3 : 2
45-45-90° triangle — 1 : 1 : √2
ApplicationWe need to apply these ratios a few times:
CD : CE = √3 : 2 ⇒ CE = (7√3)(2/√3) = 14
CE : BE = 1 : √2 ⇒ BE = 14(√2) = 14√2
BE : AE = √3 : 2 ⇒ AE = (14√2)(2/√3) = (28√2)/√3
The fraction can be simplified by multiplying by (√3)/(√3):
((28√2)/√3)×(√3)/(√3) = (28√6)/3
AE = (28/3)√6
How many four digit numbers a are there such as half of the number a is divisible by 2, a third of a is divisible by 3, and a fifth of a is divisible by 5
We have lcm(2, 3, 5) = 30, but none of 30/2 = 15, 30/3 = 10, nor 30/5 = 6 are divisible by 2, 3, and 5, respectively.
To account for this, take the square of 30, 30² = 900. Then 900/2 = 450, 900/3 = 300, and 900/5 = 180 are respectively divisible by 2, 3, and 5.
Multiply this by 2 to get a 4-digit number. It follows that [tex]\boxed{a=1800}[/tex].
The two-way table shows the number of students in a school who like hockey and/or football. like hockey do not like hockey total like football 50 25 75 do not like football 30 45 75 total 80 70 150
5 students like hockey more than football.
What is a table?A table is a collection of information or data that is commonly organized in rows and columns but can sometimes have a more sophisticated structure. Tables are common in communication, research, and data analysis. Tables can be found in a variety of media, including print media, handwritten notes, computer software, architectural ornamentation, traffic signs, and many more locations. The specific conventions and vocabulary used to describe tables differ depending on the situation. Tables also range greatly in terms of variety, structure, flexibility, nomenclature, representation, and use.To find out How many more students like hocket than football:
Refer to the given table -
Students who do not like football and like hockey = a = 30
Students who like football and do not like hockey = b = 25
Now,
[tex]=a-b\\=30-25\\=5[/tex]
Therefore, 5 students like hockey more than football.
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The question you are looking for is shown below:
The two-way table shows the number of students in a school who like hockey and/or football. like hockey does not like hockey total like football 50 25 75 do not like football 30 45 75 total 80 70 150.
How many more students like hocket than football?
How many gallons of pure water must be added to 500 gallons of a 40% saline solution to reduce it to 25% saline solution
Answer:
300 gallons
Step-by-step explanation:
A saline solution is a solution of salt in water.
The percent is the percent of salt in the total amount of solution.
A 40% saline solution has 40% of salt out of the total volume.
Let the amount of pure water needed = x.
The amount of existing 40% solution is 500 gal.
Let the total amount of 25% saline solution made = y.
Equation of volumes of solutions:
500 + x = y
y = x + 500 Eq. 1
Equation of volume of salt:
500 gal of 40% saline solution has 0.4 × 500 gal = 200 gal of salt
Pure water has 0% salt.
The final product is a solution that is 25% saline. Its volume is y.
The volume of salt in the y gallons of 25% saline solution is 25% of y = 0.25y
The equation of salt content is:
200 + 0 = 0.25y
y = 800 Eq. 2
Eq. 1 and Eq. 2 form a system of equations.
y = x + 500
y = 800
Substitute 800 for y in Eq. 1.
y = x + 500
800 = x + 500
x = 300
Answer: 300 gallons of pure water
The slant height and radius of a cone are 4 and 1, respectively. Unrolling the curved surface gives a circular sector with center angle $n^\circ.$ Find $n.$
The centre angle of circular sector is n = 90 degrees (approximately).
What is circular sector?A sector is referred to as a component of a circular made up of the circle's arc and its two radii. It is a section of the circle made up of the arc's circumference and the radius of the circle at its ends.
A piece of pizzas can be used as an analogy for the form of a circle's sector.
The formula for finding the angle formed at the circular sector in radian is;
Let 'n' be the centre angle.
Let 'l' be the slant height of the cone which is equal to radius of circular sector.
Let 'r' be the radius of the cone.
First, calculate the total length of the circular sector say 'L' which is equal to the circumference of the circular base of the cone.
circumference = 2[tex]\pi[/tex]r
= 2×3.14×1
= 6.28
Now,
centre angle = length of circular sector/radius of circle
n = L/l
n = 6.28/4
n = 1.57 radian
Convert radian in degree as;
n = (1.57×180)/[tex]\pi[/tex]
n = 89.95
n = 90 degrees (approximately)
Therefore, the angle made by the circular sector is 90 dergees.
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The correct question is-
The slant height and radius of a cone are 4 and 1, respectively. Unrolling the curved surface gives a circular sector with center angle n degrees. Find n.
What is the volume of the cube below?
Answer:
27
Step-by-step explanation:
volume = side³
v = 3³
v = 27
i dont know how to convert to whatever h is so this is the best answer i can give
Based on the data in this two-way table, which statement is true?
Using the probability concept, the correct statement is:
B. P(hibiscus|red) = P(hibiscus).
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
For item a, the probabilities are:
P(yellow|rose) = 45/105 = 0.4286.P(yellow) = 135/315 = 0.4286.Same probabilities, hence the statement that they are different is false.
For item b, the probabilities are:
P(hibiscus|red) = 80/120 = 2/3.P(hibiscus) = 210/315 = 2/3.Equal, hence this is the correct statement.
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PLEASE HELP! 20 Points!!
Shawn tried to define a reflection.
• For any point R on the line of reflection L, the image R’ is at the same point as R. • For any point P not on the line of reflection L, the image P’ is on the other side of L such that P and P’ are the same distance from L.
What mistake did Shawn make in his definition of a reflection?
Choose 1 answer from the options below.
The mistake that Shawn made in his definition of a reflection is that D. Shawn didn't make a mistake.
What's a reflection?It should be noted that reflection simply means ten transformation of the shape of an object.
In this case, for any point R on the line of reflection L, the image R’ is at the same point as R. Furthermore, for any point P not on the line of reflection L, the image P’ is on the other side of L such that P and P’ are the same distance from L.
Therefore, Shawn is right as no mistake was made.
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Answer:
PP' must be perpendicular to the line of reflection. Whatever the distance is between P and L, there are infinitely many points on the other side of L that are the same distance from L.
Step-by-step explanation:
During the summer, jody earns $10 per hour babysitting and $15 per hour doing yardwork. this week she worked 34 hours and earned $410. if x represents the number hours she babysat and y represents the number of hours she did yardwork, which system of equations models this situation? a. x y = 34 10x 15y = 410 b. x y = 410 10x 15y = 34 c. x y = 34 15x 10y = 410 d. x y = 410 15x 10y = 34
The option A is correct, the linear equation x + y = 34. and 10x + 15y = 410 represent the the statement.
According to the statement
we have given that the
Jody earns $10 per hour babysitting And jody earns $15 per hour doing yardwork.
And she worked 34 hours and earned $410
and we have to express in the linear equation terms.
So, For this purpose,
Let x represents the number hours she babysat
Let y represents the number of hours she did yardwork
And the equations become
x + y = 34.
10x + 15y = 410
And with the help of these linear equations we find the hours which are x and y.
So, The above written equations perfectly express the given conditions in the statement.
The option A is correct, the linear equation x + y = 34. and 10x + 15y = 410 represent the the statement.
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Order the numbers from least to greatest.
Help please….
Answer:
1/2=20/40 19/20=38/40
Step-by-step explanation:
so the first option
Select the correct answer.
When bisecting AB using string, which step best describes what comes after securing the string at point A and then setting the string length to be a little more than half
of AB?
When you are bisecting AB using a string and you have secured the strong at point A to be more than half of AB, the next step is to C. Make an arc on AB from point A and another arc from point B .
How should AB be bisected?If you are using a string, the first step is the secure that string at point A because the string will be used for the bisecting.
After the string is secured, take the length to just a little more than half of line AB.
Once that is done, use the string to make an arc on line AB from the top to bottom.
Unsecure the string and take it to point B. Once secured in point B, repeat the process done with point A.
This would bisect the line using a string.
In conclusion, option C - Make an arc on AB from point A and another arc from point B is correct.
Options for this question include:
A. Make an arc above AB from point A and another arc on AB from point B . B. Make an arc above and below AB from point A and another arc from point B. C. Make an arc on AB from point A and another arc from point B . D. Make an arc above and below AB from point A and another arc on AB from point B.Find out more on bisecting with a string at https://brainly.com/question/13870436
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Here is a list of numbers: 9.5, 4.4, 8.1, 2.9, 9.2, 3.2, 9.4, 7, 2.5 state the median.
The median of the list of numbers is 7
How to determine the median?The number is given as:
9.5, 4.4, 8.1, 2.9, 9.2, 3.2, 9.4, 7, 2.5
Rearrange the numbers in ascending order
2.5, 2.9, 3.2, 4.4, 7, 8.1, 9.2, 9.4, 9.5
The median is the middle number.
The middle number here is 7
Hence, the median of the list of numbers is 7
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At the Junior Olympics, Jacob ran the 500-yard
dash in 80 seconds. Juan's time for the same
distance was t seconds less than Jacob's.
Which expression would accurately calculate
Juan's time?
The Time taken by Jacob to run the dash race is (80 - t) seconds
How to write algebraic expressions?We are told that;
Total distance for the dash race = 500 yards
Time taken by Jacob to run the dash race = 80 seconds
Now, we are told that Juan ran the same dash race but used t less seconds than Jacob. Thus;
Time taken by Jacob to run the dash race = (80 - t) seconds
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Can someone help me please
Answer:
[tex]the \: answer \: is \: choice \: b \: \frac{ - 2 \sin(x) }{1 + \cos(x) } [/tex]
will give you an explanation if you want tag me on comment.
If Triangle DEF is congruent to Triangle ABC and AB=4.4 units, what is the length of DE?
2.2 units
6.6 units
8.8 units
4.4 units
Answer:
8.8
Step-by-step explanation:
If DEF///ABC
AB=4.4
2. Find the 20th term of an arithmetic sequence if its 6th term is 14 and 14th term is 6.
can anyone answer this ?
Answer:
[tex]\sf t_{20}= 0[/tex]
Step-by-step explanation:
Arithmetic sequence:[tex]\sf \boxed{\bf n^{th} \ term = a + (n-1)d}\\\\\text{Here, a is the first term ; d is the common difference }[/tex]
6th term is 14 ⇒ [tex]\sf t_6 = 14[/tex]
a + (6 - 1)d = 14
a + 5d = 14 --------------(I)
14th term is 6 ⇒[tex]\sf t_{14} = 6[/tex]
a + (14-1)d = 6
a + 13d = 6 ----------------(II)
Subtract equation (II) from equation(I)
(I) a + 5d = 14
(II) a + 13d = 6
- - -
-8d = 8
d = 8 ÷(-8)
[tex]\sf \boxed{\bf d= (-1)}[/tex]
Plugin d = -1 in equation (I)
a + 5(-1) = 14
a -5 = 14
a = 14 + 5
[tex]\sf \boxed{\bf a = 19}[/tex]
20th term:
[tex]\sf t_{20}= 19 + 19*(-1)[/tex]
= 19 - 19
[tex]\sf \boxed{\bf t_{20} = 0}[/tex]
Answer:
0
Step-by-step explanation:
The number of terms of an Arithmetic progressions has the formular.
Tn = a + ( n - 1 ) d
From the question,
6th term = 14
14th term = 6
Therefore,
a + 5d = 14 -----------(1)
a + 13d = 6 ----------(2)
subtracting
-8d = 8
dividing bothsides by -8
[tex] \frac{ - 8d}{ - 8} = \frac{8}{ - 8} \\ d = - 1[/tex]
Therefore,
common difference= -1
substituting the value of d into equation (1)
a + 5 ( -1) = 14
a - 5 = 14
a = 14 + 5 = 19
First term = 19
For the 20th term
T 20 = a + 19d
19 + 19 ( -1 )
19-19 = 0
Therefore,
20th term = 0
Please help me I really need it summer school is as
Answer:
x>-2
Step-by-step explanation:
x>-2 is the answer since we are looking for x values that the graph takes.
The line x=-2 looks like an asymptote so x>-2 must be the answer
Answer: x>-2
Step-by-step explanation:
Pleaseeeee helppp asap
Answer:
Step-by-step explanation:
Answer:
35.7 megabytes
Step-by-step explanation:
Let's first find 10% of the percent needed. If it is 234 megabytes and 10% has downloaded so far, we conclude that 10% of the download is 23.4 megabytes (since we move the decimal point one spot to the left to find ten percent).
To get to 15% of the percent needed, let's split 10% in half to get the amount of megabytes per every 5%.
23.4/2 = 11.7
We can now conclude that 5% = 11.7. Therefore, 15% will equal (23.4 + 11.7) 35.1 megabytes.
To find the remaining 3.5% left of the percent needed, we use a calculator.
3.5% x 18 = 0.63.
We will add that to the 35.1 megabytes, but we need to round to the nearest tenth, as the question stated. Therefore, we round to 0.6, add it to 35.1, to get a total of 35.7 megabytes.
For 18.5% of the download, it will be 35.7 megabytes.
(3.2x10 to the power of 5)(5.7x10 to the power of -2)
Answer:
1.320000
2. 0.057
Step-by-step explanation:
please help find the answer
Answer:
[tex]AC = \bf 4.13 \space\ cm[/tex]
Step-by-step explanation:
Using Pythagoras's theorem:
[tex]\boxed {a^2 = b^2 + c^2}[/tex],
where:
a ⇒ hypotenuse (AC, ? cm)
b, c ⇒ the two other sides of the triangle (AB = 2.2 cm, BC = 3.5 cm),
we can find the length of AC.
[tex]AC^2 = AB^2 + BC^2[/tex]
⇒ [tex]AC^2 = (2.2)^2 + (3.5)^2[/tex]
⇒ [tex]AC = \sqrt{(2.2)^2 + ((3.5)^2}[/tex]
⇒ [tex]AC = \sqrt{17.9}[/tex]
⇒ [tex]AC = \bf 4.13 \space\ cm[/tex]
The gradient of the curve y = ax² + bx at the point (3, 3) is 4. Find the value of a and the value of b.
The curve passes through the point (3, 3), so [tex]y=3[/tex] when [tex]x=3[/tex]. Then
[tex]y = ax^2 + bx \implies 3 = 9a + 3b \implies 3a + b = 1[/tex]
The tangent line to the curve at (3, 3) has gradient [tex]\frac{dy}{dx}[/tex] at [tex]x=3[/tex]. Compute the derivative.
[tex]y = ax^2 + bx \implies \dfrac{dy}{dx} = 2ax + b[/tex]
Then when [tex]x=3[/tex], the gradient is 4, so
[tex]2ax + b = 4 \implies 6a+b=4[/tex]
Solve for [tex]a[/tex] and [tex]b[/tex]. Eliminating [tex]b[/tex], we find
[tex](6a+b) - (3a+b) = 4-1 \implies 3a = 3 \implies \boxed{a=1}[/tex]
and it follows that
[tex]3 + b = 1 \implies \boxed{b = -2}[/tex]