Main Answer:The area of the right triangle is:4.5
Supporting Question and Answer:
How can we find the slope of the tangent line to a curve at a given point?
To find the slope of the tangent line to a curve at a given point, we can calculate the derivative of the function representing the curve and evaluate it at the x-coordinate of the given point. The resulting value will give us the slope of the tangent line at that point.
Body of the Solution:To find the area of the right triangle formed by the tangent line and the coordinate axes, we need to determine the length of the legs of the triangle.
Given that the tangent line is drawn to the graph of y = 4x/(1 + x^3) at the point (1, 2), we can find the slope of the tangent line at that point by taking the derivative of the function y with respect to x and evaluating it at x = 1.
Let's calculate the derivative:
dy/dx = [(1 + x^3)(d/dx)(4x) - 4x(d/dx)(1 + x^3)] / (1 + x^3)^2
= (4 + 4x^3 - 12x^3) / (1 + x^3)^2
Evaluating the derivative at x = 1:
dy/dx = (4 + 4(1)^3 - 12(1)^3) / (1 + (1)^3)^2
= (4 + 4 - 12) / (1 + 1)^2
=-4/4
=-1
Therefore, the slope of the tangent line at the point (1, 2) is -1.
Now, we can find the equation of the tangent line using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Plugging in (x1, y1) = (1, 2) and m = -1:
y - 2 = (-1)(x - 1)
y - 2 = -x +1
y = -x + 3
To find the points where this line intersects the coordinate axes, we set y or x to zero:
For y-axis (x = 0):
0 = -x + 3
x = 3
For x-axis (y = 0):
y = -0+3
Y=3
We have three points: (0, 3), (3, 0), and (1, 2), which form a right triangle.
The lengths of the legs of the right triangle can be calculated using the distance formula. Let's calculate the lengths:
Length of the vertical leg = distance between (0, 3) and (0, 0) = |3 - 0| = 3 Length of the horizontal leg = distance between (3, 0) and (0, 0)
= |3 - 0| = 3
Now, we can calculate the area of the right triangle using the formula: Area = (1/2) * base * height.
Area = (1/2) * (3) * (3)
=9/2=4.5
Hence, the area of the right triangle formed by the tangent line and the coordinate axes is 4.5.
Final Answer:Therefore, the area of the right triangle formed by the tangent line and the coordinate axes is 4.5.
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The area of the right triangle is : 4.5
How can we find the slope of the tangent line to a curve at a given point?To find the slope of the tangent line to a curve at a given point, we can calculate the derivative of the function representing the curve and evaluate it at the x-coordinate of the given point. The resulting value will give us the slope of the tangent line at that point.
To find the area of the right triangle formed by the tangent line and the coordinate axes, we need to determine the length of the legs of the triangle.
Given that the tangent line is drawn to the graph of y = 4x/(1 + x^3) at the point (1, 2), we can find the slope of the tangent line at that point by taking the derivative of the function y with respect to x and evaluating it at x = 1.
Let's calculate the derivative:
dy/dx = [(1 + x^3)(d/dx)(4x) - 4x(d/dx)(1 + x^3)] / (1 + x^3)^2
= (4 + 4x^3 - 12x^3) / (1 + x^3)^2
Evaluating the derivative at x = 1:
dy/dx = (4 + 4(1)^3 - 12(1)^3) / (1 + (1)^3)^2
= (4 + 4 - 12) / (1 + 1)^2
=-4/4
=-1
Therefore, the slope of the tangent line at the point (1, 2) is -1.
Now, we can find the equation of the tangent line using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Plugging in (x1, y1) = (1, 2) and m = -1:
y - 2 = (-1)(x - 1)
y - 2 = -x +1
y = -x + 3
To find the points where this line intersects the coordinate axes, we set y or x to zero:
For y-axis (x = 0):
0 = -x + 3
x = 3
For x-axis (y = 0):
y = -0+3
Y=3
We have three points: (0, 3), (3, 0), and (1, 2), which form a right triangle.
The lengths of the legs of the right triangle can be calculated using the distance formula. Let's calculate the lengths:
Length of the vertical leg = distance between (0, 3) and (0, 0) = |3 - 0| = 3 Length of the horizontal leg = distance between (3, 0) and (0, 0)
= |3 - 0| = 3
Now, we can calculate the area of the right triangle using the formula: Area = (1/2) * base * height.
Area = (1/2) * (3) * (3)
=9/2=4.5
Hence, the area of the right triangle formed by the tangent line and the coordinate axes is 4.5.
Therefore, the area of the right triangle formed by the tangent line and the coordinate axes is 4.5.
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The image of the point (8,1) under a translation is (9,2). Find the coordinates of
the image of the point (5,5) under the same translation.
1-2-3-4-5-6-7-8-9-10
0.10)/%
Barry wants to make a drawing that is 1/2 the size of the original. If a tree in the original drawing is 11 inches tall and 5 inches wide,
Answer:
Barry's drawing is 5.5 in tall and 2.5 in wide.
Step-by-step explanation:
We are given that
Height of tree, h=11 in
Width of tree, b=5 in
We have to find the size of the original.
Size of drawing is 1/2 size of the original
Therefore,
Height of drawing, h'=[tex]\frac{1}{2}\times 11[/tex]in
Height of drawing, h'=5.5 in
Width of drawing tree=[tex]\frac{1}{2}\times 5[/tex]
Width of drawing tree=[tex]2.5in[/tex]
Hence, Barry's drawing is 5.5 in tall and 2.5 in wide.
Please help me 3x+y=-3
Answer:
x=0 y=-3
Step-by-step explanation:
please help me on this will give you brainliest
Answer:
57.603
Step-by-step explanation:
We can separate this into two sections, using the and:
Fifty-seven - simply the whole number 57
Six hundred three thousandths - This could be intrepreted as 0.603, which is confirmed through the "3" being in the thousandths place.
Sorry that took so long! I had to think a bit about this one
Brainliest need help plz
A horizontal line has points A , E, D. A line extends vertically from point E to point C and forms a right angle at C E D. A line extends up and to the left from point E to point B. Which statement is true about the given information? ∠CED measures 45°. ∠CED measures 180°. ∠AEC measures 90°. ∠AEC measures 45°.
Answer:
The answer is 90
Step-by-step explanation: Im not 100% sure but yeah.
What is greater 100,000 cm or 1 mi PLEASE HELLP AND PLEASE SHOW YOUR WORK QUICK
Answer:
yeah 100,000 cm is greater
Step-by-step explanation:
What statement is true for life insurance coverage?
O A. The policy only covers beneficiary's future needs, funeral costs should not be included in the
policy amount
B. Debt is forgiven at death, so insurance policies do not need to cover this cost.
O
C Life insurance policy amounts should factor in how much it would take to maintain the standard
of living for the beneficiaries
D. Generally speaking, $10,000 is the recommended amount for a life insurance policy
Answer:
C: Life insurance policy amounts should factor in how much it would take to maintain the standard of living for the beneficiaries
Step-by-step explanation:
The statement is true for life insurance coverage
Life insurance policy amounts should factor in how much it would take to maintain the standard of living for the beneficiaries.
What is Life insurance coverage?A contract between an insurance policy holder and an insurance company in which the insurer agrees to pay a quantity of money in exchange for a premium upon the demise of an insured person or after a specific amount of time is known as life insurance.
In order to provide a death benefit in exchange for paying premiums, life insurance functions. One common kind of life insurance, called term life insurance, only lasts for a predetermined period, such 10 or 20 years. Although the death benefit of permanent life insurance likewise lasts during the policyholder's lifetime as long as premiums are paid, it does not.
Thus, the true statement is
Life insurance policy amounts should factor in how much it would take to maintain the standard of living for the beneficiaries.
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For 3/4 of a pound of peanuts, Jimmy paid $2.35. What is the unit rate (dollars per pound)? *
Answer:
$3.13/lb
Step-by-step explanation:
Divide 3/4 pound into $2.35:
$2.35
--------- = $3.13/lb
3/4 lb
brainliest to whoever gets it right !
Cándido normally pays $875 for bodily injury and property damage insurance. His insurance company increases premiums by 150% for 1 accident, 200% for 2-3 accidents, and 250% for 4 accidents. Find his premium for the year after he has three accidents to the nearest dollar.
Answer:
$1,700.
Step-by-step explanation:
Since Cándido normally pays $ 875 for bodily injury and property damage insurance, and his insurance company increases premiums by 150% for 1 accident, 200% for 2-3 accidents, and 250% for 4 accidents, to find his premium for the year after he has three accidents to the nearest dollar, the following calculation must be performed:
875 x 200 / 100 = X
875 x 2 = X
1,700 = X
Therefore, his premium for the year after he has three accidents is $1,700.
Answer:
premium = $ 875
increase 200%
875 * 200 / 100
=$1750
Step-by-step explanation:
plz help!!!! i’ll mark brainliest!!
Find the unit rate for this situation: type 145 words in 5 minutes
Answer:
29 words per minute
Step-by-step explanation:
[tex] \frac{145}{5} = 29 \\ \frac{29}{1} = 29[/tex]
Answer:
29 words per minute
Step-by-step explanation:
145/5 = 29 words per minute
Hope that helps!
GameStop is going out of business and everything is 60% off. What is
the new price of a game that regularly costs $58.99?
Answer:
With original price $58.99 and 60% off,
Final price: $23.60
Saved amount: $35.39
Step-by-step explanation:
Use the graph and table to write the equation that describes the line graphed.
A 2-column table with 5 rows. Column 1 is labeled x with entries negative 2, negative 1, 0, 1, 2. Column 2 is labeled y with entries 1, 2, 3, 4, 5.
The equation of the line
Please help me :(
Answer:
I believe it is x−y+3=0.
Step-by-step explanation:
y= mx + b
The equation of a line is written as [tex]\frac{y_{2-}{y_{1} } }{x_{2-}{x_{1} } }[/tex], so if you plug in the numbers from the first two rows on the grid you get, [tex]\frac{2-1}{-1--2}[/tex] = [tex]\frac{1}{1}[/tex] = 1
The y- intercept is [tex]b= y_{1}[/tex]-m·[tex]x_{1}[/tex], plug in the number and its, b=1-(1)·(-2)=3
Finally, the equation of the line can be written in the form y=mx+b.
y=x+3.
The equation of the line in the slope-intercept form is y=x+3.
The equation of the line in the point-slope form is y−1=1(x+2).
The general equation of the line is x−y+3=0.
Answer:
It’s D y=X+3
Step-by-step explanation:
La energía que posee un cuerpo que se encuentra en movimiento se conoce como energía cinética. La Física es la ciencia encargada de estudiar este fenómeno natural y, para calcularla, utiliza la fórmula. La unidad de medida de la energía es el julio (J).
Answer:
Energía cinética = 0,5 x (masa x velocidad^2).
Step-by-step explanation:
"Cinética" significa "movimiento", y por lo tanto, la energía cinética es la energía que hay en algún objeto o cuerpo porque se está moviendo. Así, esta energía puede definirse como el trabajo mecánico necesario para acelerar un objeto de masa particular desde el carril hasta el terreno en movimiento. Al recibir esta energía rápidamente, el cuerpo retiene esta energía cinética dentro de él, a menos que cambie su velocidad.
????????????????????????????
Answer:
Pretty sure it's c)
What the radical form of 13 2/3
Answer:
1/13 2/3
Step-by-step explanation:
The total circular area a sprinkler can water is 240 2 . The sprinkler is set to rotate back and forth in a 75° pattern. Find the amount of area that will get watered.
Answer:
The sprinkler will cover an area of [tex]50m^2[/tex]
Step-by-step explanation:
Given
[tex]Area = 240m^2[/tex]
[tex]\theta = 75^{\circ}[/tex]
Required
Determine the amount of area it can cover
The amount of area is calculated by calculating the area of a sector using:
[tex]Sector\ Area =\frac{\theta}{360^{\circ}} * \pi r^2[/tex]
Where:
[tex]Area = \pi r^2 = 240m^2[/tex]
So, we have:
[tex]Sector\ Area =\frac{\theta}{360^{\circ}} * 240m^2[/tex]
Substitute 75 for [tex]\theta[/tex]
[tex]Sector\ Area =\frac{75^{\circ}}{360^{\circ}} * 240m^2[/tex]
[tex]Sector\ Area =\frac{75^{\circ}* 240m^2}{360^{\circ}}[/tex]
[tex]Sector\ Area =\frac{75* 240m^2}{360}[/tex]
[tex]Sector\ Area =\frac{18000m^2}{360}[/tex]
[tex]Sector\ Area =\frac{18000}{360} m^2[/tex]
[tex]Sector\ Area =50m^2[/tex]
Hence, the sprinkler will cover an area of [tex]50m^2[/tex]
I do not understand this at all
Answer:
2s+3b=154
s=2b
solving: s= 44, b= 22
Step-by-step explanation:
s=number of sundaes sold
b=number of banana splits sold
2s+3b=154
s=2b
to solve:
substitute s=2b into the first equation
2(2b)+3b=154
4b+3b=154
7b=154
divide both sides by 7
7b/7=154/7
b=22
s=44
The winning times for a race are shown in the table.
Which answer describes the rate of change from year 2 to year 4?
- The winning time increased by an average of 0.267 second per year from year 2 to year 4.
- The winning time increased by an average of 0.8 second per year from year 2 to year 4.
- The winning time increased by an average of 0.4 second per year from year 2 to year 4.
- The winning time increased by an average of 0.475 second per year from year 2 to year 4.
Answer: The winning time increased by an average of 0.4 second per year from year 2 to year 4.
The winning time for a race displayed in the table would be:
C). The winning time increased by an average of 0.4 seconds per year from year 2 to year 4.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Find the increase in winning time:
Given that,
Year Winning Time(seconds)
1 27.5
2 28.4
3 28.7
4 29.2
5 29.44
Hence, The total change in the time is,
= (0. + 0.3 + 0.5 + 0.24)/4
= 0.485 seconds
Now, If we round off, it becomes 0.5 seconds.
Therefore, it can be considered that an average of 0.5 seconds changes every year from year 2 to 4.
Thus, ''Option C'' is the correct answer.
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The width of a rectangle is 7 inches and the length is 13 longer than the width. Find the perimeter of the rectangle. (PLEASE ANSWER QUICKLY!!!)
HELP! Please answer fast How many terms are in the geometric series 0.4 + 1.2 + 3.6 + ... + 7,873.2?
Answer:
10 terms. I think you should check the explanation :)
Step-by-step explanation:
0.4 + 1.2 + 3.6 + ... + 7,873.2?
So I figured that you have to multiply the terms by three to get the next term.
0.4 * 3 = 1.2
1.2 * 3 = 3.6
But, I know it will takes us a really long time to get to 7,873.2, so what I will do is jump 3 sequences.
So if I multiply by 3 to get the next term, I'll multiply by 9 to get the next two terms because 3 * 3 is 9.
So to jump three times, I will multiply by 27 to get the next three terms because 3 * 3 * 3 is 27.
0.4 + 1.2 + 3.6 + ...
3.6 * 27 = 97.2
0.4 + 1.2 + 3.6 + ..., ..., 97.2, ...
97.2 * 27 = 2624.4
0.4 + 1.2 + 3.6 + ..., ..., 97.2, ..., ..., 2624.4, ...
Since we are getting close, I'm going to go back to multiply by 3.
2624.3 * 3 = 7873.2
0.4 + 1.2 + 3.6 + ..., ..., 97.2, ..., ..., 2624.4, 7873.2
Now that we have reached 7873.2, we will now count the terms (count the sets of three dots as terms too).
There are 10 terms in the geometric sequence.
What unit should be used to calculating fundraising for field trip
Answer:
the mcps i think
Step-by-step explanation:
Im not sure :/
What is the simplified form of this expression? (-3x2 + x + 5) − (4x2 − 2x) A. 7x2 + 3x − 5 B. -7x2 + 3x + 5 C. X2 − x + 5 D. -x2 + 3x − 11
Show set of data that is a function using at least 5 domain values and 5 range values. It can be shown in a format of your choice
Answer:
Five domain values of the function f(x) = x² - 1, are; 1, 2, 3, 4, 5
Five range values of the function f(x) = x² - 1, are; 0, 3, 8, 15, 24
Step-by-step explanation:
The values in the domain are the values of the independent or (normally) x-variable
The values i the range are the values obtained by applying the function relationship to the domain as follows;
Let 'f(x)' represent the function that generates the values of the data in the data set,
For f(x) = x² - 1, we have the following table;
[tex]\begin{array}{ccc}x&f(x)\\1&0\\2&3\\3&8\\4&15\\5&24\end{array}[/tex]
Therefore;
The function is f(x) = x² - 1
The five domain values are; 1, 2, 3, 4, 5
The five range values are; 0, 3, 8, 15, 24.
3 What is the value of x2 + 4 for x = 5? -))
Answer:
14
Step-by-step explanation:
substitute 5 for x
5(2) + 4 = 10 + 4 = 14
Andre's recipe for lemonade calls for 5 cups of
sugar to make 3 gallons of lemonade. Leila will
follow the same recipe to make g gallons of
lemonade using s cups of sugar. Which of these
equations represents the relationship between g
Answer: [tex]g= \frac{5}{3} s[/tex]
Step-by-step explanation:
[tex]3g=5s\\g=\frac{3}{5}s[/tex]
Susie bought 6 yards of fabric to make scarves. She estimates that it takes 3/5 yard to make huge scarf. How many scarves can Susie make?
jake scored 2 goals in soccer this season. jamie scored 4 times as many goals as jake. Edgar scored 3 times as many goals as jamie. How many goals did Edgar score?
Answer:
24
Step-by-step explanation:
2x4 = 8 8x3 = 24 so edgar has 24 goals
Answer: 24
Step-by-step explanation:
Jake scored 2
Jamie scored 4x2=8
Edgar scored 8x3=24
re-write the equation in Ax+By=C form. use integers for A,B, and C.
y+2=-5(x-5)
Answer:
5x+y=23
Step-by-step explanation:
y+2=-5(x-5)
y+2=-5x+25
y=-5x+25-2
y=-5x+23
23=y-(-5x)
23=y+5x
5x+y=23
solve for x. -3(x+5)=-9
Answer:
x = -2
Step-by-step explanation: