Differentiate the function/equation with respect to x and solve for the derivative, dy/dx. The value of dy/dx at the given point is the slope of the tangent line to the curve at that point. Then use the point-slope formula to get the equation of the tangent.
1.
[tex]y = \dfrac{x^2}{x+2} \implies \dfrac{dy}{dx} = \dfrac{2x\times(x+2) - x\times1}{(x+2)^2} = \dfrac{x(x+4)}{(x+2)^2}[/tex]
When x = 2, the derivative is
[tex]\dfrac{dy}{dx}\bigg|_{x=2} = \dfrac{2(2+4)}{(2+2)^2} = \dfrac34[/tex]
Then the equation of the tangent line at (2, 1) is
[tex]y - 1 = \dfrac34 (x - 2) \implies \boxed{y = \dfrac{3x}4 - \dfrac12}[/tex]
2.
[tex]x^3 + 2y^2 = 10y \implies 3x^2 + 4y \dfrac{dy}{dx} = 10 \dfrac{dy}{dx} \implies \dfrac{dy}{dx} = \dfrac{3x^2}{10-4y}[/tex]
When x = 2 and y = 1, the derivative is
[tex]\dfrac{dy}{dx}\bigg|_{(x,y)=(2,1)} = \dfrac{3\times2^2}{10-4\times1} = 2[/tex]
Then the tangent at (2, 1) has equation
[tex]y - 1 = 2 (x - 2) \implies \boxed{y = 2x - 3}[/tex]
Factor the quadratic equation of y= 4x^2 + 16x -48
Answer:
y = 4(x+6)(x-2)
Step-by-step explanation:
y = 4x² + 16x - 48, factor out the 4
y = 4(x² + 4x - 12), find factors of 12
Factors of 12:
1*12, 2*6, 3*4; find a pair that sums to 4x or 4
6-2 = 4, since 12 is a negative number, a number in the factor must be negative, in this case, making 2 negative, would make 6-2 = 4
y = 4(x+6)(x-2)
Given the graph, find the initial value of the domain [ ___, 2].
Answer:
Looks to me like it should be -2
A pyramid has a square base with sides of length s. The height of the pyramid is equal to 1/2 of the length of a side on the base. Which formula
represents the volume of the pyramid?
A. V = 1/12 s ^3
B. V=1/6 s ^3
C. V=1/3s ^3
D. V=3s³
E. V=6s³
Answer:
The volume of the pyramid with a base area of s² and the heights iss
The formula for calculating the volume of a square based pyramid is expressed as:
Volume = 1/3* Base area* Height
Since it is a square-based pyramid, the base area is calculated as:
Base Area = s ^ 2
If the height of the pyramid is equal to of the length of a side on the base, hence h = s
Substituting the base area and the height into the formula for the volume, this will give;
V = BH / 3
V = (s ^ 2 * s)/3
V = (s ^ 3)/3
V = 1/3 * s ^ 3
Hence the volume of the pyramid with a base area of s ^ 2 and the height s is 1/3 * s ^ 3
Gertrude has a life insurance policy that will pay her family $97,000 per year I she dies. If interest rates are at 1.4% when the insurance company has to pay, what is the amount of the lump sum that the insurance company must put into a bank account? A. $692,857,142.90 B. $6,928,571.43 c. $692,857.14 D. $69,285,714.29
The amount of the lump sum that the insurance company must put into a bank account is $6,928,571.43 option (B) is correct.
What is the percentage?It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.
We have:
Gertrude has a life insurance policy that will pay her family $97,000 per year.
If interest rates are at 1.4% when the insurance company has to pay
r = 1.4%
or
r = 0.014
The amount of the lump sum that the insurance company must put into a bank account:
= 97,000/0.014
= 6928571.429 ≈ $6,928,571.43
Thus, the amount of the lump sum that the insurance company must put into a bank account is $6,928,571.43 option (B) is correct.
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Solve x^2 - 10x = 30x
[tex]x^2-10x=30x\\x^2-40x=0\\x(x-40)=0\\x=0 \vee x=40[/tex]
PLEASEEE HELPPP!!
Using the sheet music example above, please name the parts:
________________________
________________________
What is this LINE of music called? ________________________
When both of these staves are connected with this symbol, what is this called? ________________________
Please name all 3 things inside of the box: ___________________________________
_________________________
The line of music is refered to as the bracket
What is music?It should be noted that music simply means a vocal or instrumental sounds that are combined to produce beauty of harmony and expression if emotions.
When When both of these staves are connected with this symbol, it's called a brace.
Lastly, the things inside the box are voice, piano, and guitar.
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Which numbers below are odd?
A. 4271
B. 7966
C. 787
D. 288
E. 8113
F. 985
The points (2,3) and (5,9) are on the line. What is the equation of the line
The equation of the line is y = 3x +3 .
What is a Straight Line Equation ?A straight line equation is that can be represented by y = mx +c , where m is the slope and c is the intercept on y axis.
The points given are (2,3) and (5,9)
m = (9-3)/(5-2) = 3
m = 3
Therefore the equation becomes
y = 3x +c
Te equation is given by
(y-y₁) = m(x-x₁)
y-3 = 3(x - 2)
y-3 = 3x -6
y = 3x +3
Therefore the equation of the line is y = 3x +3 .
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What is the volume of the complex figure
The volume of the given figure is 104 m³.
We need to find the volume of the given figure.
What is volume?Volume is a scalar quantity expressing the amount of three-dimensional space enclosed by a closed surface.
Divide the given figure into two convenient cuboids. That is one with dimensions length=4 m, breadth=4 m and height=3 m. Another with length=4 m, breadth=7 m and height=2 m.
We know that, the volume of cuboid =l×b×h cubic units.
Now, the volume of cuboid=4×4×3=48 m³ and the volume of another
cuboid=4×7×2=56 m³.
The volume of the given figure=48+56=104 m³.
Therefore, the volume of the given figure is 104 m³.
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Immediate help needed please.
Can you answer and explain please
The possible values of k are k < - 1.868 or k > 0.535 in which the inequality is true.
What is a quadratic equation?Any equation of the form [tex]\rm ax^2+bx+c=0[/tex] where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.
As we know, the formula for the roots of the quadratic equation is given by:
[tex]\rm x = \dfrac{-b \pm\sqrt{b^2-4ac}}{2a}[/tex]
We have a quadratic equation:
(k - 1)x² + (4k)x + k -3 = 0
Here,
a = k - 1
b = 4k
c = k - 3
As we know for distinct real roots:
D > 0
[tex]\rm {b^2-4ac}} > 0[/tex]
(4k)² - 4(k-1)(k-3) > 0
[tex]\rm 16k^2-4\left(k-1\right)\left(k-3\right) > 0[/tex]
[tex]\rm 12k^2+16k-12 > 0[/tex]
[tex]\rm 3k^2+4k-3 > 0[/tex]
[tex]\rm 3\left(k+\dfrac{2}{3}\right)^2-\dfrac{13}{3} > 0[/tex]
[tex]\rm 3\left(k+\dfrac{2}{3}\right)^2 > \dfrac{13}{3}[/tex]
[tex]\rm \left(k+\dfrac{2}{3}\right)^2 > \dfrac{13}{9}[/tex]
[tex]\rm k < \dfrac{-\sqrt{13}-2}{3}\quad \mathrm{or}\quad \:k > \dfrac{\sqrt{13}-2}{3}[/tex]
or
[tex]\rm k < -1.868 \quad \mathrm{or}\quad \:k > 0.535[/tex]
Thus, the possible values of k are k < - 1.868 or k > 0.535 in which the inequality is true.
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Mrs. Avery is going to randomly select one student from her class to read a poem out loud. There are 151515 boys and 131313 girls in her class.
What is \text{P(boy})P(boy)start text, P, left parenthesis, b, o, y, end text, right parenthesis?
If necessary, round your answer to 222 decimal places.
The probability that a boy is selected is 0.5357 or 53.57%
What is the probability?Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.
P(boy) is asking for the probability that a boy will be chosen to read a poem out loud. Since there are 15 boys, we can get the probability by dividing this by the total number of students.
This is because out of all the students, there is an equal possibility that each of the 15 boys will be chosen.
P = ( 15 ) / ( 15+13)
P = 15 / 28
P = 0.5357
Therefore the probability that a boy is selected is 0.5357 or 53.57%
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Answer:
53.57%
Step-by-step explanation:
Kmele had paid his credit card on time for three years, but last month he missed the due date. He then noticed that the amount owed increased by $35. This most likely happened because he was charged.
A. a late fee
B. a higher interest rate
C. a minimum-payment fee
D. an overdraft-fee
Answer:
It's most likely he was charged a late fee
what is π x 5
please tell me the answer as quick
Please help me!!! I will give brainliest to the correct answer.
[tex]\mathsf {2(10 + 2) = 3(y + 3)}[/tex]
[tex]\mathsf {24 = 3y + 9}[/tex]
[tex]\mathsf {3y = 15}[/tex]
[tex]\textsf {y = 5}[/tex]
[tex]\textsf {The value of y will be 5. }[/tex]
Identify parallel and perpendicular lines
Parallel lines are those on a plane that do not intersect while perpendicular lines are those that intersect at an angle of 90°.
How are parallel and perpendicular lines different?Parallel lines will never intersect with each other and will maintain the same distance from each other for as long as they are on the same plane.
Perpendicular lines on the other hand will intersect each other and will do so at a 90° angle. An example of such a line is the steps on a ladder which intersect the beam at a 90° angle.
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The function f(x)=5(x−1) is shown on the coordinate plane.
Select from the drop-down menus to correctly describe the end behavior of f(x).
As x decreases without bound, the graph of f(x) ______
1. approaches y=0
2. increases without bond
3. decreases without bond
As x increases without bound, the graph of f(x) ______
1. approaches y=0
2. increases without bond
3. decreases without bond
As x decreases without bound, the graph of f(x) approaches y = 0
As x increases without bound, the graph of f(x) increases without bound
What is Function?An expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given:
f(x)=5^(x−1)
When x -> -∞
[tex]lim_{ x- > -\infty.[/tex] f(x)= 5 ^(x-1) = 0
When x -> ∞
[tex]lim_{ x- > \infty.[/tex] f(x)= 5 ^(x-1) = ∞
Hence,
As x decreases without bound, the graph of f(x) approaches y = 0
As x increases without bound, the graph of f(x) increases without bound
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What is the length of CD? In this diagram, AABC~ AEDC.
The length of CD will be option C. 3
From given diagram, triangle ABC and EDC meet at a common vertex C.
So, triangle ABC and EDC both are congruent to each other.
Applying the similarity property of congruence, ratio of the congruent triangles' corresponding sides will be equal.
Therefore,
AB/ED = AC/EC = BC/CD
AC/EC = BC/CD
18/6 = (12 - x)/x
Simplifying the expression,
We get
18x = 6(12 - x)
Multiplicating the terms in right hand side,
18x = 72 - 6x
Solving 6x on both sides,
we get
18x + 6x = 72 - 6x + 6x
24x = 72
Dividing both the sides by 24, we get
24x/24 = 72/24
x = 3
Hence, the length of CD is 3.
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a skyscraper casts a shadow 200 feet long. If the angle of elevation of the sun is 38 degrees, then the height of the skyscraper is approximately__.
A. 173.34 feet
B. 156.26 feet
C. 300 feet
D. There is not enough information.
144% of what number is 54?
Answer:
37.5
Step-by-step explanation:
37.5×144/100=54 the number is 37.5
Solve the following system of equations:
4x − 2y − z = −5
x − 3y + 2z = 3
3x + y − 2z = −5
(0, 1, 3)
(0, −1, 3)
(0, 1, −3)
(0, −1, −3)
Answer: (0, 1, 3)
Step-by-step explanation:
Adding the second and third equations, we get [tex]4x-2y=-2[/tex]Multiplying both sides of the first equation by 2, we get that [tex]8x-4y-2z=-10[/tex]. Adding this to the second equation, we get that [tex]9x-7y=-7[/tex].If we multiply both sides of the equation [tex]4x-2y=-2[/tex] by 7, we get [tex]28x-14y=-14[/tex].
If we multiply both sides of the equation [tex]9x-7y=-7[/tex] by 2, we get [tex]18x-14y=-14[/tex].
Subtracting the equation [tex]28x-14y=-14[/tex] from the equation [tex]18x-14y=-14[/tex], we get that [tex]-10x=0 \longrightarrow x=0[/tex].
This means that [tex]4(0)-2y=-2 \longrightarrow y=1[/tex]
And thus, [tex]4(0)-2(1)-z=-5 \longrightarrow -2-z=-5 \longrightarrow z=3[/tex].
So, the solution is (0, 1, 3).
f(x)=2(x)^2+5√(x+2)
f(2)=
Answer:
f(2) = 18
Step-by-step explanation:
Put '2' into the equation where 'x' is to get :
f(2) = 2 (2^2) + 5 (sqrt(2+2)
= 8 + 5 (2) = 18
En una tienda se fabrican dos tipos de mesas, de 3 y 4 patas. En total se tienen 89 mesas para vender. Se sabe que el quíntuple de las mesas de 3 patas más la cantidad de mesas de 4 patas es igual a 277. ¿ Cuántas mesas de 3 patas se tienen para vender?
En una tienda se fabrican
La cantidad de mesas de 3 patas que tiene para la venta la tienda que las
fabrica es: 47
¿Qué es un sistema de ecuaciones?
Es un arreglo de ecuaciones que se caracteriza por tener el mismo número de incógnitas que de ecuaciones.
Existen diferentes métodos para su resolución:
Sustitución: se despeja de una ecuación una variable, quedando en función de otra para luego sustituirla en otra ecuación y así obtener
Igualación: se despeja la misma variable en dos de las ecuaciones y se igualan los resultados.
Eliminación: se resta o suman dos ecuaciones para que quede un resultado en función una variable y así despejarla.
Gráfico: se grafican las rectas y el punto de intersección es la solución del sistema.
¿Cuántas mesas de 3 patas se tienen para vender?
Definir;
x: mesas de 3 patas
y: mesas de 4 patas
Ecuaciones
x + y = 89
5x + y = 277
Aplicar método de eliminación;
Restar 1 - 2;
x + y = 89
-5x - y = -277
-4x + 0 = -188
Despejar x;
x = 188/4
x = 47
Sustituir;
47 + y = 89
Despejar y;
y = 89 - 47
y = 42
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Simplify this problem!!!
Answer:
C
Step-by-step explanation:
HOPE THAT HELPSBDNQBDB
Which expression is equivalent to 2^5?
Answer:
the equivalent fractions for 2/5 are:
(2/5) × (2/2) = (2 × 2) / (5 × 2) = 4/10
(2/5) × (3/3) = (2 × 3) / (5× 3) = 6/15
(2/5) × (4/4) = (2 × 4) / (5 × 4) = 8/20
Answer:
4/10, 6/15, 8/20, etc.
Step-by-step explanation:
I hope you got some idea
There are 60 students studying Sciences (Biology, Chemistry or Physics). 7 students study Biology, Chemistry and Physics. 12 students study Biology and Chemistry. 15 students study Biology and Physics. 20 students study Chemistry and Physics. 35 students study Biology. 32 students study Physics. B Complete the Venn diagram. E P C There are 60 students studying Sciences ( Biology , Chemistry or Physics ) . 7 students study Biology , Chemistry and Physics . 12 students study Biology and Chemistry . 15 students study Biology and Physics . 20 students study Chemistry and Physics . 35 students study Biology . 32 students study Physics . B Complete the Venn diagram . E P C
Answer:
8 in the box.
Step-by-step explanation:
As, it is given that 35 students study Biology.
So, in Biology circle,
15+7+5+x=35
27+x=35
x=35-27
x=8
how do i work this out?
She has 45% of the original amount left
Ratio and proportionsFractions are written as a ratio of two integers
Given the following
Initial amount. = ∈4000
Amount given to her sister = 1/4 * 4000 = 1000
Amount given to her brother = 40% of 3000 = 1200
Amount left = 4000 - (1000+1200)
Amount left = ∈1800
Determine the percentage left
x * 4000 = 1800
x = 1800/4000
x = 0.45
x = 45%
Hence she has 45% of the original amount left
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solve the equation uding the most direct method: 3x(x+6)=-10?
To solve this problem, you will use the distributive property to create an equation that can be rearranged and solved using the quadratic formula.
DistributeUse the distributive property to distribute 3x into the term (x + 6):
[tex]3x(x+6)=-10[/tex]
[tex]3x^2+18x=-10[/tex]
RearrangeTo create a quadratic equation, add 10 to both sides of the equation:
[tex]3x^2+18x+10=-10+10[/tex]
[tex]3x^2+18x+10=0[/tex]
Use the Quadratic FormulaThe quadratic formula is defined as:
[tex]\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
The model of a quadratic equation is defined as ax² + bx + c = 0. This can be related to our equation.
Therefore:
a = 3b = 18c = 10Set up the quadratic formula:
[tex]\displaystyle x=\frac{-18 \pm \sqrt{(18)^2 - 4(3)(10)}}{2(3)}[/tex]
Simplify by using BPEMDAS, which is an acronym for the order of operations:
Brackets
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Use BPEMDAS:
[tex]\displaystyle x=\frac{-18 \pm \sqrt{324 - 120}}{6}[/tex]
Simplify the radicand:
[tex]\displaystyle x=\frac{-18 \pm \sqrt{204}}{6}[/tex]
Create a factor tree for 204:
204 - 1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102 and 204.
The largest factor group that creates a perfect square is 4 × 51. Therefore, turn 204 into 4 × 51:
[tex]\sqrt{4\times51}[/tex]
Then, using the Product Property of Square Roots, break this into two radicands:
[tex]\sqrt{4} \times \sqrt{51}[/tex]
Since 4 is a perfect square, it can be evaluated:
[tex]2 \times \sqrt{51}[/tex]
To simplify further for easier reading, remove the multiplication symbol:
[tex]2\sqrt{51}[/tex]
Then, substitute for the quadratic formula:
[tex]\displaystyle x=\frac{-18 \pm 2\sqrt{51}}{6}[/tex]
This gives us a combined root, which we should separate to make things easier on ourselves.
Separate the RootsSeparate the roots at the plus-minus symbol:
[tex]\displaystyle x=\frac{-18 + 2\sqrt{51}}{6}[/tex]
[tex]\displaystyle x=\frac{-18 - 2\sqrt{51}}{6}[/tex]
Then, simplify the numerator of the roots by factoring 2 out:
[tex]\displaystyle x=\frac{2(-9 + \sqrt{51})}{6}[/tex]
[tex]\displaystyle x=\frac{2(-9 - \sqrt{51})}{6}[/tex]
Then, simplify the fraction by reducing 2/6 to 1/3:
[tex]\boxed{\displaystyle x=\frac{-9 + \sqrt{51}}{3}}[/tex]
[tex]\boxed{\displaystyle x=\frac{-9 - \sqrt{51}}{3}}[/tex]
The final answer to this problem is:
[tex]\displaystyle x=\frac{-9 + \sqrt{51}}{3}[/tex]
[tex]\displaystyle x=\frac{-9 - \sqrt{51}}{3}[/tex]
math questionawdAawdadaDaWD
Substitute 5 in the place of X
[tex]f(x) = \frac{1}{8} \times 4 ^{x} \\ f(5) = \frac{1}{8} \times 4 ^{5} \\ f(5) = \frac{1}{8} \times 1024 \\ f(5) = 0.125 \times 1024 \\ f(5) = 128[/tex]
The asnwer is D
PLEASE GIVE BRAINLIEST
3(2x - 3) = 4(2x + 4)
Answer:
Step-by-step explanation:
Open the brackets
3 x 2x = 6x
3x -3 = -9
4 x 2x= 8x
4 x 4= 16
6x - 9 = 8x + 16
Move same to same side
-16 -9 = 8x - 6x
-25 = 2x
x -25/2
x = -12.5
Answer:
x = -25/2 or -12.5Step-by-step explanation:
3*(2x - 3) = 4*(2x + 4)
6x - 9 = 8x + 16
6x - 8x = 16 + 9
-2x = 25
x = -25/2 or -12.5
------------------------
check
3 * (2 * (-12.5) - 3) = 4 * (2 * (-12.5) + 4)
3 * (25 - 3) = 4 * (12.5 + 4)
3 * 22 = 4 * 16.5
66 = 66
the answer is good
If you receive a 15% MARKUP on an item costing $60.00, how much money do you pay in total?
Answer:
The total money we have to pay is $69.
Step-by-step explanation:
MARKUP is the amount added to the cost price.
MARKUP formula = 100*profit / cost
Cost of the item = $60.00
put the value in the formula
15 = 100*profit/60
profit = 60 × 15 ÷ 100 = 9
the total money is cost + profit
which is 60 + 9 = $69.
So the total money we have to pay is $69.
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