The linear approximation of the function is given by a³ - 3a² +4 + 10ax
The given function is f(x)=x³+5x²+4
Let us find the linear approximation.
L(a) = f(a) + f'(a)(x-a)
let us differentiate the function
f'(x) = 2x²+10x
Let us find the value of the function at the exact point a
f'(a) = 2a²+10a
f(a) = a³+5a²+4
L(a) = f(a) + f'(a)(x-a)
L(a) = a³ + 5a² + 4 + 2a² + 10a (x-a)
L(a) = a³ + 7a² +4 + 10ax - 10a²
L(a) = a³ - 3a² +4 + 10ax
The use of a line to approximate the value of a function at a point is the definition of a linear approximation of either a function. When we see a curve (of a function) and a point on it, we immediately think of the tangent line concept.
If the equation of the tangent line is discovered at the supplied point, one can roughly calculate the value of something similar to the function at every location that is very close to the supplied point using the equation of the tangent line. This concept is also known as the tangent line approximation because the tangent line is utilized. Additionally called linear approximation.
Therefore the linear approximation of the function is given by
a³ - 3a² +4 + 10ax .
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help meeeeeeeeeeeeeee pleaseeeeeee
Answer: 19.6 feet
Step-by-step explanation:
Using the Pythagorean theorem,
[tex]x^2 +(x+6)^2 =48^2\\\\x^2 +x^2 +12x+36=2304\\\\2x^2 +12x-2268=0\\\\x^2 +6x-1134=0\\\\x=\frac{-6 \pm \sqrt{6^2 -4(1)(-1134)}}{2(1)}\\\\x \approx 30.8 \text{ } (x > 0)\\\\\implies x+(x+6) \approx 67.6\\\\\therefore (x+(x+6))-48 \approx 19.6[/tex]
What is the sum of -12, 5, and -10?
0-27
0-17
O 17
O 27
Answer:
The answer is -17
Step-by-step explanation:
Add all the numbers together:
-12-10 is -22
Then -22+5= -17
the measure of two angle are (8x+24) and (12x-16) what is the value of x if theses angle ar congruent
Answer: 10
Step-by-step explanation: Congruent means they are the same, so you set up the problem as 8x+24=12x-16. Then you just solve for x. Subtract 8x from both sides to get 24=4x-16. Then add 16 to both sides. 40=4x. Then divide by 4 into both sides to get x=10.
help meeeeeeeeeeeeeee pleaseeeeeee
Answer: 2.5, 5.4
Step-by-step explanation:
[tex]-16t^2 +126t=213\\\\16t^2 -126t+213=0\\\\t =\frac{-(-126) \pm \sqrt{(-126)^2 -4(16)(213)}}{2(16)}\\\\t \approx 2.5, 5.4[/tex]
The diagram below shows a pennant in the shape of an isosceles triangle. The equal sides each measure 13, the altitude is x+7, and the base is 2x. 13 x + 7 2x 13 What is the length of the base? Your answer
we have that
the altitude divide the isosceles triangle into two congruent right triangles
so
Applying the Pythagorean Theorem
(x+7)^2=13^2+x^2
solve for x
x^2+14x+49=169+x^2
simplify
14x+49=169
14x=169-49
14x=120
x=120/14
simplify
x=60/7
Find the length of the base
2x=2(60/7)=120/7
answer is 120/7 unitsit takes james 15 minutes to cut a log into 4 pieces. find how long it will take him to cut a log into 15 pieces.
It will take 56.25 minutes to cut the log into 15 pieces.
What is an equation?Two or more expressions with an equal sign are defined as an equation.
Given that, James takes 15 minutes to cut the log into 4 pieces, which can be represented by:
4 pieces = 15 minutes
1 piece = 15/4 minutes
15 pieces = (15/4)×15 minutes
15 pieces = 56.25
Hence, it will take 56.25 minutes to cut the log into 15 pieces.
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4. Felix and Oscar applied for the same credit card from the same bank.
The bank checked both of their FICO scores. Felix had an excellent
credit rating, and Oscar had a poor credit rating.
a. Felix was given a card with an APR of 12%. What was his
monthly percentage rate?
b. Oscar was given a card with an APR of 15%. What was his
monthly payment?
c. If each of them had an average daily balance of $800 and had to
pay a finance charge, how much more would Oscar pay than Felix?
A) The monthly percentage rate of Felix is; 1%
B) The monthly percentage rate of Oscar is; 1.25%
C) The amount that Oscar will pay more than Felix is; $2
What is the monthly Percentage Rate?A monthly interest rate is simply how much percentage interest you would be charged in one month while annual percentage interest rate is defined as the percentage interest rate you will be charged annually.
A) We are told that Felix was given a card with APR of 12%.
Thus;
Monthly percentage rate = 12%/12 = 1%
B) We are told that Oscar was given a card with an APR of 15%. Thus;
Monthly percentage rate = 15%/12 = 1.25%
C) We are told that each of them had an average daily balance of $800 for a specific month.
Thus;
Amount paid by Felix = 800 * 1%
Amount paid by Felix = $8
Amount paid by Oscar = 800 * 1.25%
Amount paid by Oscar = $10
Difference in Payments = $10 - $8 = $2
Thus, Oscar pays $0.96 more than Felix.
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8. Ethan earns $13 per hour at his341job. Last week he workedhours. How much did he earn lastweek?A $13.75C $445.25B $442.00D $448.50
To solve the exercise, first it is convenient to convert the mixed number to decimal:
[tex]34\frac{1}{2}=34+\frac{1}{2}=34+0.5=34.5[/tex]Now, you can use a rule of three:
[tex]\begin{gathered} 1\text{ hour}\rightarrow\text{\$13} \\ 34.5\text{ hours}\rightarrow\text{\$x} \end{gathered}[/tex][tex]\begin{gathered} x=\frac{34.5\text{ hours }\cdot\text{ \$13}}{1\text{ hour}} \\ x=\frac{34.5\cdot\text{ \$13}}{1} \\ x=34.5\cdot\text{ \$13} \\ x=\text{ \$}448.5 \end{gathered}[/tex]Therefore, Ethan did earn $ 448.50 last week.
Solve for g: g + 8 > 10
Given the inequality:
[tex]g+8>10[/tex]Solve for g, subtract 8 from both sides
[tex]\begin{gathered} g+8-8>10-8 \\ \\ g>2 \end{gathered}[/tex]so, the answer will be:
[tex]g=(2,\infty)[/tex]Cooling towers are used to remove or expel heat from a process. A cooling tower's walls are modeled by x squared over 400 minus quantity y minus 110 end quantity squared over 2304 equals 1 comma where the measurements are in meters. What is the width of the cooling tower at the base of the structure? Round your answer to the nearest whole number. please help look at the image
100 meters
50 meters
48 meters
40 meters
The width of the cooling tower at the base of the structure is 40 meters.
What is defined as the ellipse?An ellipse is a closed curve made up of points for whom distances from two points (foci) each add up to a single value. The center is the midpoint between the foci. One property of the an ellipse that the reflection of a line from one focus off its boundary will pass across the other.For the given question,
The equation modelling the cooling tower's walls is -
x²/400 - (y-110)²/2,304 = 1
This can be written as;
x²/20² - (y-110)²/48² = 1
Comparing the equation with the standard form of ellipse;
(x-h)²/a² - (y-k)²/b² = 1
The base of the ellipse will be its major axis,
That is 2a.
Thus,
For the width of the colling tower.
Cooling tower width =2√400
Cooling tower width = 2×20
Cooling tower width= 40 meters
Thus, the width of the cooling tower at the base of the structure is 40 meters.
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Sydney measured a line to be 4.9 inches long. If the actual length of the line is 5.2 inches, then what was the percent error of the measurement, to the nearest tenth of a percent?
The percent error of the measurement will be 5.8%.
What is mean by Percentage?
A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.
To Calculate the percent of a number , divide the number by whole number and multiply by 100.
Given that;
Sydney measured a line to be 4.9 inches long.
And, The actual length of the line is 5.2 inches.
Now,
The percent error is calculated as;
The difference between the measures = 5.2 - 4.9
= 0.3 inches
So, The percent error = 0.3/5.2 x 100
= 5.76%
Thus, The percent error of the measurement to the nearest tenth of a percent will be 5.8%.
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I dont really understand this. (10th grade math)
Answer:
Angle 1: 56 degrees
Angles 2: 56 degrees
Angles 3: 74 degrees
Step-by-step explanation:
triangles angles added up equal 180
So we add the 2 angles in the first triangle to find angle 1
58+66
124
Then subtract by 180
180-124
56
Angle 1 is 56 degrees
angles 2 is also 56 degrees since they are corresponding angles
Now we know 2 angles of the second triangle and to find angle 3 we add the 2 angles
56+50=106
And subtract by 180
180-106
74
Angles 3 is 74 degrees
Hopes this helps please mark brainliest
Just help with the problem and do it quickly please
SOLUTION
In 2020 black engineers under 30 is 70% of 500, which is
[tex]\frac{70}{100}\times500=350[/tex]In 2021 500 members joined making a total of 1000 members.
The black engineers together are now 50% of 1000, which is 500
So the numbers of black engineers that joined in 2021 becomes
[tex]500-350=150[/tex]Hence the answer is 150
on a number line, which rational number is between 23.07 and 24.11
C) 24.10
1) Considering that rational numbers are any number that can be written as a ratio.
The number 23.07 is the same as 2307/100 and
The number 24.11 is the same as 2411/100
If we take the number line, we have a rational number 24.10 between those.
Because 24.10 can be written as 2410/100
Show work please!
29) Write two expressions that are equivalent to model below.
30)Write two expressions that are equivalent to the model below.
31) Write an expression to represent the model below.
32) One cotton candy at the rodeo cost $5.25. When 4 friends purchases cotton candy, the expression 5.25(4) can be used to determine the total cost. What other expression can be used to determine the total cost?
29. The two equivalent expressions for the models are: 3x - 6 and 3(x - 2).
30. 2x + 2 and 2(x + 1).
31. 36x.
32. Another expression for the total cost is: 4(5.25).
What are Equivalent Expressions?Expressions are said to be equivalent expressions when both of them have identical terms, or both gives the same value if the values of the variables in both expressions are plugged in.
29. In the model, we have 3 x's, which is: x + x + x = 3x.
We also have 6 "-1". Which is: 6 × -1 = -6.
An expression for the model would be: 3x - 6
It can also be rewritten as: 3(x - 2).
The two equivalent expressions are: 3x - 6 and 3(x - 2).
30. We have, 2x - 4 + 6 = 2x + 2
It can be rewritten as 2(x + 1).
The two equivalent expressions are: 2x + 2 and 2(x + 1).
31. Let each of the shapes be represented as x. They are 36 in number. Therefore, an expression for the model would be 36x.
32. Another expression that can be used to determine the total cost is: 4(5.25).
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(c-d2) = 7d (if c = 11 and d = 5)
Which exponential function models this situation ?
Exponential function models is v(x) = 9,000(0.9)ˣ
Lets take two year data (0,9000) and (1,8100)
using the general form of exponential function y = abˣ and substitute value of x and y as (0,9000)
9000 = ab⁰
9000 = a.1
9000 = a
Again using general form of exponential function and taking x and y as (1 , 8100)
8100 = ab¹
substitute a = 9000 in above equation
8100 = 9000×b
b = 8100/9000
b=0.9
putting value of a and b in general form of exponential function
v(x) = 9000(0.9)ˣ is required exponential function model
correct answer is option B
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Aline has a slope of 2, and passes through the point (6,8). Determine which equation in slope intercept form) represents this line.
the equation is of the form y = mx + b, where m = 2, then
for b:
[tex]\begin{gathered} y=mx+b \\ 8=2(6)+b \\ 8=12+b \\ 8-12=12+b-12 \\ b=-4 \end{gathered}[/tex]equation is
[tex]y=2x-4[/tex]answer: C.
Determine if the proportion is true 1/2= 3/4 Proportion is true Proportion is not true
K:
1/2 and 3/4 are NOT equivalent.
If you divide 1 by 2, the result is 0.5
If you divide 3 by 4, the result is 0.75
Therefore, they are NOT equivalent
1/2 and 2/4 are equivalent:
If you divide 1 by 2, the result is 0.5
If you divide 2 by 4, the result is also 0.5
Now, you can answer the question.
What is the equation of the line that passes through the point (-8,-8)(−8,−8) and has a slope of 22?
The slope intercept form of the required line is 2x - y = -8
What is equation of line in slope intercept form?
The most general form of equation of line in slope intercept form is given by y = mx + c
Where m is the slope of the line and c is the y intercept of the line.
Slope of a line is the tangent of the angle that the line makes with the positive direction of x axis.
If [tex]\theta[/tex] is the angle that the line makes with the positive direction of x axis, then slope (m) is given by
m = [tex]tan\theta[/tex]
The distance from the origin to the point where the line cuts the x axis is the x intercept of the line.
The distance from the origin to the point where the line cuts the y axis is the y intercept of the line.
Slope of the line = 2
The line passes through (-8, -8)
Equation of the required line-
[tex]y - (-8)= 2(x - (-8))\\y + 8 = 2(x + 8)\\y + 8 = 2x + 16\\2x - y = 8 -16\\2x - y = -8[/tex]
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.
Tyler owns a food truck that sells chicken wings and hamburger sliders. Each order of
chicken wings sells for $7.25-while each order of hamburger sliders sells for $3. On
Friday, Tyler made a total of $595 in revenue from all sliders and wings-sales. There
were twice as many chicken wing orders sold than there were sliders sold.
The number of chicken wing orders is 90 and the number of sliders is 45.
According to the question,
We have the following information:
Each order of chicken wings sells for $7.25-while each order of hamburger sliders sells for $3. On Friday, Tyler made a total of $595 in revenue from all sliders and wings-sales. There were twice as many chicken wing orders sold than there were sliders sold.
Now, let's take the number of sliders to be x and the number of chicken wing orders to be y.
y = 2x
Now, we have the following expression:
7.25x+3(2x) = 595
7.25x+6x = 595
13.25x = 595
x = 45 (This value is rounded off to a whole number)
Now, the number of wing sales:
y = 2*45
y = 90
Hence, the number of chicken wing orders is 90 and the number of sliders is 45.
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pinpoint a the diameter is 12 CD is 8 and MCD is 90 find each measure round to the nearest hundred
The value of mDE =45, F.D=4, A.F=8
A circle is a spherical shape that lacks corners and edges. A circle is a two-dimensional, closed curved shape.
Given that diameter is 12 cd =8 and mcd =90
We have to find mDE =?, F.D, A.F
From the given diagram
Diameter = BE = 12
CD =8 so C.F=D.F=4
Let EF = x then B.F =12- x
BY properties of circle C.F × FD = B.F x EF
4 x 4 = x(12-x)
X2 – 12x + 16 =0
Because BE is the diameter and CD is the perpendicular, we can conclude that arcs CE and DE are equal.
arc(CD) = arc(CE) + arc(DE)
90 = 2.arc(CE)
arc(CE) = 45 = arc(DE)
A.F= 12 – 4 =8
Therefore the value of mDE =45, F.D=4, A.F=8
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Answer:
The value of mDE =45, F.D=4, A.F=8
A circle is a spherical shape that lacks corners and edges. A circle is a two-dimensional, closed curved shape.
Step-by-step explanation:
Determine if each expression is equal to (2) (-3) (-4)(6) (-4)(-3) (8)(12) (2)(-4) (-6)
The product of the numbers (2) (-3) (-4) is:
[tex]2\times-3\times-4=24[/tex]By verifying the options one after the other, to check which is also equal to 24, we have:
[tex](6)(-4)=-24[/tex][tex](-3)(8)=-24[/tex][tex](12)(2)=24[/tex][tex](-4)(-6)=24[/tex]Hence, the correct options are C and D
What is the missing value in the proportion 4/4=x/30 A.6B.24C.37.5D.29
Answer
x = 30
Explanation
We need to find the value of x in the proportion
[tex]\begin{gathered} \frac{4}{4}=\frac{x}{30} \\ \text{Cross multiply} \\ 4x=120 \\ \text{Divide both sides by 4} \\ \frac{4x}{4}=\frac{120}{4} \\ x=30 \end{gathered}[/tex]Hope this Helps!!!
6 chairs cost £84 in total.
How much does 1 chair cost?
Give your answer in pounds (£).
Answer: £14
Step-by-step explanation:
Divide the pounds from the chairs, in this case £84 / 6 is £14. So each chair costs £14.
how do I find the value of xfor the following triangles
Given at the figure 1
A right triangle of legs of 10 and 16
We need to find the angle x
The side of 16 represent the opposite side of the angle x
The side of 10 represents the adjacent side of the angle x
According to the known sides, we can find x using tan function :
So,
[tex]\begin{gathered} \tan x=\frac{opposite}{adjacent} \\ \\ \tan x=\frac{16}{10}=1.6 \\ \\ x=\tan ^{-1}1.6=57.995 \end{gathered}[/tex]Rounding to the nearest degree
So, the measure of angle x = 58
the altitude of a triangle is increasing at a rate of 1 1 centimeters/minute while the area of the triangle is increasing at a rate of 4 4 square centimeters/minute. at what rate is the base of the triangle changing when the altitude is 10.5 10.5 centimeters and the area is 90 90 square centimeters?
Answer:
-128/147 cm/min
Step-by-step explanation:
[tex]A=\frac{1}{2}bh \\ \\ 90=\frac{1}{2}(10.5)b \\ \\ b=\frac{120}{7} \\ \\ \\ A=\frac{1}{2} bh \\ \\ \implies \frac{dA}{dt}=\frac{b}{2} \frac{dh}{dt}+\frac{h}{2} \frac{db}{dt} \\ \\ 4=\frac{21}{4}+\frac{db}{dt}+\frac{60}{7} \\ \\ \frac{db}{dt}=-\frac{128}{147}[/tex]
Which expression is equivalent to the following complex fraction?
11
-IX-X
411X11
11
0
y
y+X
y-x
(y-x)(y+x)
x²y²
x²y²
(y+x)(y+x)
y-x
y+X
The equivalent expression of the given complex fraction is (5x - 2y)/(3x - 2y).
We are given a fraction. The fraction is complex in nature. A fraction is a portion of a whole or, more broadly, any number of equal parts. A complex fraction is a rational expression that contains a fraction in either the numerator or the denominator. Let the complex fraction be represented by the variable "F". The fraction is given below.
F = [(-2/x) + (5/y)]/[(3/y) - (2/x)]
We have to convert this complex fraction into a simple mathematical expression. A mathematical expression is a finite collection of symbols that is well-formed according to context-specific norms. Let the expression be represented by the variable "E".
E = [(-2y + 5x)/x*y]/[(3x - 2y)/x*y)]
E = [(-2y + 5x)/(3x - 2y)]/[x*y/x*y)]
E = [(5x - 2y)/(3x - 2y)]*1
E = (5x - 2y)/(3x - 2y)
The complete question is given below.
Which expression is equivalent to the following complex fraction?
[(-2/x) + (5/y)]/[(3/y) - (2/x)]
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Please help! Help is much appreciated. I think I already got a.
b.
To determine how many accounts will minimize the long run average cost.
Given the function;
[tex]C=576-48N+N^2[/tex]We need to plot a graph of the function and identify the lowest point on the graph ( since we are minimizing).
Above is a section of the graph of the function.
The lowest point on the graph is at;
[tex]\begin{gathered} N=24 \\ C=0 \end{gathered}[/tex]Therefore, the number of accounts that will minimize the long run average cost is 24 accounts.
Also, the minimum long run average cost is 0
c.
We want to determine whether the territories should be merged together.
To do this we will check their average long run cost when they merge and compare to their average long run cost when they are separate;
When the merge, the total number of accounts is;
[tex]N=16+19=35[/tex]So, the long run average cost is;
[tex]\begin{gathered} C=576-48N+N^2 \\ C=576-48(35)+35^2 \\ C=121 \end{gathered}[/tex]When they are separate, as calculated in a.
the long run average cost for both is;
[tex]\begin{gathered} C_1=576-48(16)+16^2 \\ C_1=64 \\ C_2=576-48(19)+19^2 \\ C_2=25 \\ C=C_1+C_2=64+25 \\ C=89 \end{gathered}[/tex]So, since the long run average cost for when they merge is greater that the total long run average cost when the are separate, then they should not merge the two territories because merging the two territories increases the long run average cost. And we all know the aim is to always try to reduce cost.
Therefore, they should not merge
The height of a triangle is 3 feet less than the base. The area of the triangle is 230 square feet. Find the length of the baseand the height of the triangle,
Let b be the base of the traingle
Let h be the height of the triangle
The height of a triangle is 3 feet less than the base:
[tex]h=b-3[/tex]The area of the triangle is 230 square feet.
The area of a triangle is:
[tex]A=\frac{1}{2}(b\cdot h)[/tex]For the given triangle:
[tex]\begin{gathered} A=\frac{1}{2}(b\cdot(b-3)) \\ \\ 230=\frac{1}{2}(b\cdot(b-3)) \end{gathered}[/tex]Solve b in the equation above:
[tex]\begin{gathered} 230=\frac{b^2-3b}{2} \\ \\ 2\cdot230=b^2-3b \\ \\ 460=b^2-3b \\ \\ b^2-3b-460=0 \end{gathered}[/tex]Use the quadratic formula:
[tex]\begin{gathered} ax^2+bx+c=0 \\ \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \\ \end{gathered}[/tex][tex]\begin{gathered} b=\frac{-(-3)\pm\sqrt[]{(-3)^2-4(1)(-460)}}{2(1)} \\ \\ b=\frac{3\pm\sqrt[]{9+1840}}{2} \\ \\ b=\frac{3\pm\sqrt[]{1849}}{2} \\ \\ b=\frac{3\pm43}{2} \\ \\ b_1=\frac{3+43}{2}=\frac{46}{2}=23 \\ \\ b_2=\frac{3-43}{2}=-\frac{40}{2}=-20 \end{gathered}[/tex]As the length of the base cannot be a negative quantity you use the solution 1.
The base of the triangle is 23ft
Use the value of b to find the heigth:
[tex]\begin{gathered} h=b-3 \\ h=23-3 \\ h=20 \end{gathered}[/tex]Then, the given triangle has the next dimensions:base: 23ftheight: 20ft