The equation of the line that is normal to the curve y² + ln(x) = 19 at the point (e³, -4) is given as follows:
y + 4 = -8e³(x - e³).
How to obtain the equation of the line normal to the curve?The equation of the line normal to a curve follows the point-slope definition of a linear function, given as follows:
y - y' = m(x - x').
The derivative of the function is found applying implicit differentiation, as follows:
2y dy/dx + 1/x = 0
dy/dx = -1/(2xy).
At x = e³, y = -4, the numeric value of the derivative is given as follows:
dy/dx = e^(-3)/8.
The normal line is perpendicular to the tangent line, hence it's slope is calculated as follows:
m x e^(-3)/8 = -1.
m = -8e³.
Then the equation of the line normal to the curve at the point is given as follows:
y + 4 = -8e³(x - e³).
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Terri is sawing wood into shorter pieces of equal lengths. Some original pieces are 30 inches long, some are 48 inches long, and some are 72 inches. She cannot waste any wood. What is the longest possible length of the shorter pieces that Terri saws?
Answer:
24 inches
Step-by-step explanation:
To find the longest possible length of the shorter pieces that Terri can saw, you need to find the greatest common factor (GCF) of the three original lengths: 30 inches, 48 inches, and 72 inches.
The GCF of 30, 48, and 72 can be found by finding the prime factorization of each number and taking the lowest exponent for each prime factor.
30 = 2 * 3 * 5
48 = 2^4 * 3
72 = 2^3 * 3^2
The GCF is 2^3 * 3 * 5, or 24 inches. Therefore, the longest possible length of the shorter pieces that Terri can saw is 24 inches.
Hello,
I hope you and your family are doing well!
To find the GCF is to use the Euclidean algorithm, which involves repeatedly dividing the larger of the two numbers by the smaller one until the remainder is 0. This can be a quicker method for finding the GCF of larger numbers.
For example:
72 / 30 = 2 with a remainder of 12
30 / 12 = 2 with a remainder of 6
12 / 6 = 2 with a remainder of 0
The last non-zero remainder is 6, so the GCF is 6.
Regardless of the method used, the GCF of 30, 48, and 72 is 6, so the longest possible length of the shorter pieces that Terri can saw is 6 inches.
---
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Happy Holidays!
It takes Eddie 2 hours to shovel the snow from a driveway and sidewalk. Brad can shovel the snow from the same driveway and sidewalk in 5 hours. How long would it take them working together to shovel the snow?
How do I solve it?
Eddie and Brad together shovel the snow in 1 hour 26 minutes.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given that it takes Eddie 2 hours to shovel the snow from a driveway and sidewalk.
Brad can shovel the snow from the same driveway and sidewalk in 5 hours.
We now find how long would it take them working together to shovel the snow.
Eddie does 1/2 in 1 hour or x/2 in x hours
Brad does 1/5 in 1 hour or x/5 in x hours
the sum is 1 full job
(x/2)+(x/5)=1
multiply both sides by 10
7x=10
x=(10/7) hours or 600/7 minutes =86 minutes
As we know 1 hour has 60 minutes.
We can write 86 minutes as 1 hour and 26 minutes.
Hence, 1 hour 26 minutes is the time taken by Eddie and Brad to shovel the snow.
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Please help
Whats
30-19+29+30-29-10=
Answer:
31Step-by-step explanation:
[tex]\tt 30-19+29+30-29-10[/tex]
[tex]\tt 11+29+30-29-10[/tex]
[tex]\tt 40+30-29-10[/tex]
[tex]\tt 70-29-10[/tex]
[tex]\tt 41-10[/tex]
[tex]\tt 31[/tex]
__________________
Hope this helps you!
Have a nice day!
For 3, use the table at the right.
3. Jose has $100 to spend at the pet store. He needs to buy
1 bag of dog food and 2 chew toys. How many catnip toys
can Jose buy? Write equations to show each step.
DATA
Barky's Pet Store
Product
Bag of Dog Food
Tex
Bag of Cat Food
Tex
Chew Toy
Catnip Toy
Cost
$35
$18
$12
39
The number of catnip toys that Jose can buy will be 3 catnip toys. All the steps are described below.
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The table is given below.
Product Cost
Bag of Dog Food $35
Bag of Cat Food $18
Chew Toy $12
Catnip Toy $9
In the event that, your numbers are unique, you should simply supplant the qualities you have with the methodology here, and you ought to find an exact solution.
Presently, Emma has $100 to spend at the store. Along these lines, we really want to ascertain the amount she spends on shopping:
$35 + 2 x $12 = $59
In this way, she burned through $59 in the store, so she has left:
$100 - $59 = $41
She has $41 left to spend on something different. How about we check whether she has sufficient means to purchase multiple times catnip toys as bite toys:
Chew toys: 3 x 12 = $36
Catnip toy: 3 x 9 = $27
Furthermore, Emma has $41 left, so she can either purchase multiple times bite toys or multiple times catnip toys, however not the two of them at that point, on the grounds that the all-out would be 63% dollars, and she will be missing $22.
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The formula for estimating the population variance from the sample involves taking the sum of squared deviations and dividing it by ___
The formula for estimating the population variance from the sample involves taking the sum of squared deviations and dividing it by N - 1
Population variance is a measure of dispersion that determines how far each data point is from the population mean.
On the population, the population variance is calculated. When the number of observations grows, a few data points are chosen to represent the entire population.
The variance calculated on these specific data points is referred to as the sample variance. The population variance can be estimated using the sample variance.
When the population variance is unknown , we estimate the population variance by using sample variance
The formula for estimating the population variance from the sample involves taking the sum of squared deviations and dividing it by N-1
Formula of sample variance = [tex]s^2 = \frac{\sum f_i(x_i - \bar{x})}{N-1}[/tex]
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An independent group of food service personnel conducted a survey on tipping practices in a large metropolitan area. They collected information on the percentage of the bill left as a tip for 20 randomly selected bills. The average tip was 11.1% of the bill with a standard deviation of 2.8%. Assume that the tips are approximately normally distributed. Construct an interval to estimate the true average tip (as a percent of the bill) with 90% confidence. Round the endpoints to two decimal places, if necessary.
Assuming that the tips are approximately normally distributed, an interval to estimate the true average tip (as a percent of the bill) with 90% confidence is (10.47, 11.73).
How to construct the confidence interval?Assuming that the tips are approximately normally distributed, we would determine both the lower limit and upper limit for the confidence interval where 90% of the sample means occur as follows;
Lower limit = μ - σ/√x
Lower limit = 11.1 - 2.8/√20
Lower limit = 11.1 - 0.63
Lower limit = 10.47
For the upper limit, we have:
Upper limit = μ - σ/√x
Upper limit = 11.1 + 2.8/√20
Upper limit = 11.1 + 0.63
Upper limit = 11.73
For 90% of the sample means to occur, the confidence interval is as follows;
Confidence interval = (10.47, 11.73)
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The graph shows a translation of the function f(x) = |x|. What absolute value function is represented by the graph?
A V shape with vertex at negative 1 point 5 comma zero and passing through the points negative 2 comma 1 and negative 1 comma 1.
A. f(x) = |x| + 1.5
B. f(x) = |x| − 1.5
C. f(x) = |x − 1.5|
D. f(x) = |x + 1.5|
The absolute value function that is represented in the graph is f(x) = |x + 1.5| So the correct option is D.
What absolute value function is represented by the graph?The general absolute value function is:f(x) = |x|This absolute value function has a vertex at the point (0, 0). We know that the graphed function has a vertex at the point (-1.5, 0).
So basically we had a horizontal translation of -1.5 units or 1.5 units to the left. That is written as: g(x) = f(x + 1.5)
Replacing the absolute value function we get:g(x) = |x + 1.5|
Then the correct option is D.
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Simplify the inequality
if three sides of a triangle are in ratio of 3:5:7 and the perimeter of the triangle is 12xm. what is the length of the longest side?
The required length of the longest side will be 28/5 cm in the triangle.
What is the perimeter of a triangle?The perimeter of a triangle is defined as the addition of the lengths of the triangle's three sides.
We have been given that the three sides of a triangle are in the ratio of 3:5:7
Let's assume the sides are 3x, 5x, 7x
Since the perimeter of the triangle is 12 cm.
So, 3x+5x+7x = 12
Apply the addition operation,
15x = 12
x = 12/15
Reduce the above fraction to the simplest form
x = 4/5
So, longest side = 7x = 28/5 cm
Thus, the required length of the longest side will be 28/5 cm in the triangle.
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Can you answer 2 and 3
Answer: 2. 37.8%
3. 30% decrease
Step-by-step explanation:
Increase = New Number - Original Number
Then: divide the increase by the original number and multiply the answer by 100.
% increase = Increase ÷ Original Number × 100.
36.99-22.99=14
100(14/36.99)
37.8%
First: work out the difference (decrease) between the two numbers you are comparing.
Decrease = Original Number - New Number
Then: divide the decrease by the original number and multiply the answer by 100.
% Decrease = Decrease ÷ Original Number × 100
2.50-1.75=0.75
100(0.75/2.5)
30%
Five thousand tickets are sold at $1 each for a charity raffle. Tickets are to be drawn at random and monetary prizes awarded as follows: 1 prize of $700, 3 prizes of $100, 5 prizes of $10, and 20 prizes of $5. What is the expected value of this raffle if you buy 1 ticket?
This raffle's expected value per ticket is $ 0.25.
Given that 5,000 tickets are sold at $1 apiece for a charity raffle, and tickets are to be picked at random, the following monetary prizes will be awarded.
To estimate the expected value of this raffle if you buy 1 ticket, apply the following calculation: 1 prize of $ 500, 3 prizes of $ 200, 5 prizes of $ 10, and 20 prizes of $ 5.
⇒ (500 + 3 x 200 + 5 x 10 + 20 x 5) / 5000 = X
⇒ (500 + 600 + 50 + 100) / 5000 = X
⇒ 1250/5000 = X
Apply the division operation, and we get
⇒ 0.25 = X
Therefore, this raffle's expected value per ticket is $ 0.25.
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If two balls with different speeds roll off the edge of a table at the same time, which one hits the floor first, the faster or the slower ball? Why or why not?
Both balls will strike the ground simultaneously, but the faster ball will do so farther away from the table.
What is acceleration due to gravity ?The unit g stands for gravitational acceleration. At sea level, the standard value of g on earth's surface is 9.8 m/[tex]s^{2}[/tex]
They will strike the ground simultaneously if the table and the ground are both flat.
It is customary to divide an object's motion into a vertical component and a horizontal component. While they are on the table, neither of the balls is moving vertically. Both balls begin to accelerate downward as soon as they leave the table due to the effects of gravity. Given that both balls experience the same acceleration from gravity, they will fall at the same rate and with the same downward velocity.
Both balls will strike the ground simultaneously, but the faster ball will do so farther away from the table.
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During summer, the height of the water in Scott's pool decreases by 4 inches each week, due to evaporation. How many weeks does it take for the water to decrease by 20 inches?
Responses
A −5
B 5
C 20
D −20
Answer:
5 weeks
Step-by-step explanation:
20 inches / ( 4 inches/week ) = 5 weeks
Use the L-shaped composite figure formed by two rectangular prisms to answer the question.
What is the surface area of the composite figure in square inches?
(WILL GIVE BRAINLIEST)
The surface area of the composite figure is calculated as: 640 square inches.
What is the Surface Area of a Rectangular Prism?The surface area of a rectangular prism = 2(w*l + h*l + h*w), where:
l = length w = width h = heightThe composite figure can be decomposed into two rectangular prisms, 1 and 2.
Rectangular prism 1:
l = 10 in.
w = 0.5 ft. = 6 in.
h = 4 + 7 = 11 in.
Surface area = 2*(6*10 + 11*10 + 11*6) = 472 in.²
Rectangular prism 2:
l = 10 in.
w = 0.5 ft. = 6 in.
h = 4 in.
Surface area = 2·(6·10+4·10+4·6) = 248 in.²
The area of the surface where both meet = (l)(w) = (10)(4) = 40 in.²
Therefore:
Surface area of the composite figure = 472 + 248 - 2(40) = 640 in.².
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Find the direction of the sum of these two vectors:
By using the concept of vector, it can be calculated that
Direction of the sum is 10.87° below the horizontal.
What is vector?
Anything that has both magnitude and direction and follow the vector addition law is called vector.
A = 2.71 m
[tex]A_x = 2.71 cos 60 = 1.355 \ m\\A_y = -2.71 sin 60 = -2.35 \ m[/tex]
B = 3.14 m
[tex]B_x = 3.14 cos 30 = 2.71 \ m\\B_y = 3.14 sin 30 = 1.57 \ m[/tex]
Sum of A and B = (1.355 +2.71) i + (-2.35 + 1.57) j
= 4.065i - 0.78j
Let the angle be [tex]\theta[/tex]
[tex]tan\theta = \frac{0.78}{4.065}\\tan\theta = 0.192\\\theta = tan^{-1}(0.192)\\\theta = 10.87^{\circ}[/tex]
Direction of the sum is 10.87° below the horizontal.
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Assume that the function [tex]f(x) = ax^{2} + bx + c, a \neq 0[/tex] has two real zeros. Prove that the x-coordinate of the vertex of the graph is the average of the zeros of [tex]f[/tex].
(Hint: Use the Quadratic Formula)
Use the given measurements to solve the triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
a=400, b=300
[tex]\frac{\sin \theta}{300}=\frac{\sin 2\theta}{400} \\ \\ \frac{\sin \theta}{300}=\frac{2\sin \theta \cos \theta}{400} \\ \\ \frac{1}{300}=\frac{\cos \theta}{200} \\ \\ \cos \theta=\frac{2}{3} \\ \\ \theta=\arccos(2/3) \\ \\ \theta \approx \boxed{48^{\circ}}[/tex]
Solve using quadratic formula: 5x^2+9=4x+5
Answer:
No real solutions.
Step-by-step explanation:
No real solutions.
Three varieties of coffee Coffee A, Coffee B, and Coffee C are combined and roasted, yielding a 55-lb batch of coffee beans. Twice as many pounds of Coffee C, which retails for $10.39 per lb, are needed as Coffee A, which sells for $14.99 per lb. Coffee B retails for $13.99 per lb. How many pounds of each coffee should be used in a blend that sells for $12.61 per lb?
Coffee A is 12.241 pounds, Coffee B is 18.269 pounds and Coffee C is 24.482 pounds.
What is Algebra?A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.
Variables are the name given to these symbols because they lack set values.
In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.
Given:
Let a pounds of coffee A needed
Let b pounds of coffee B needed
Let c pounds of coffee C needed
c= 2a.................(1)
a+ b+c= 55..........(2)
and, 14.99 a+ 13.99 b + 10.39 c /55 = 12.61
1499a + 1399 b + 1039 c = 69,355...............(3)
Subtract (2) from (3), we get
1499 a+ 1399 b + 1039 c= 69355
- 1039 a - 1039 b - 1039 c = 57145
_____________________________
460 a + 360 b = 12210
46a+ 36 b = 1221
b= 1221/ 36 - 46 a/ 36
b= 407 / 12 - 23 a / 18
Put b in Equation (2)
a + 407 / 12 - 23 a / 18 + 2a= 55
-5a / 18 + 2a + 407/ 12 = 55
31a/ 18 = 253/ 12
a= 4554 / 372
a = 2277/ 186
a= 759/ 62
a= 12.241
and, c=2a = 24.482
b= 33.91 - 15.641
b= 18.269
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4 There are between 20 and 50 flags.
They can be shared equally among
2 tents or 4 tents or 5 tents.
How many flags are there?
When there are between 20 and 50 flags and they can be shared equally among 2 tents or 4 tents or 5 tents. The number of flags will be 40.
How to illustrate the information?From the information, there are between 20 and 50 flags and they can be shared equally among 2 tents or 4 tents or 5 tents.
It should be noted that this simply means that we should find the highest multiple for 2, 4, and 5 between 20 and 50. This is 40.
Therefore, the number of flags is 40.
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PLEASE best explanation gets brainliest
Answer:
55
Step-by-step explanation:
Because you would minus 180-125 to get angle 1 and they said that angle 2 is parallel so they would be the same anwser so the answer is 55.
Expand the log completely
Answer:
log(2)√xy
________
y
there good luck
Consider the simultaneous equations
3x² + 2xy = 4
x + y = a
where a is a real constant.
Find the complete set of values of a for which the equations have two distinct real solutions for x.
There are no values of a.
-2 < a < 2
-1 < a < 1
a = 0
a < -1 or a < 1
a < -2 or a < 2
All real values of a
The correct option is All real values of a.
What is the real number?
The positive and negative integers, as well as the fractions created from them (also known as rational numbers), as well as the irrational numbers, are all real numbers.
We have,
3x² + 2xy = 4 ...................(1)
x + y = a ...................(2)
By converting the second equation equals to y:
y = a - x .....................(3)
By substituting the (3)rd equation into (1), we get
3x² + 2xy = 4
3x² + 2x(a - x) = 4
3x² + 2ax - 2x² = 4
x² + 2ax = 4
x² + 2ax - 4 = 0
Discriminant = 4a² + 16 >= 0
2 distinct solutions & is ok for all a.
hence, the correct option is All real values of a.
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Find the circumference and the area of a circle with radius 9m .
Use the value 3.14 for , and do not round your answers. Be sure to include the correct units in your answers.
The circumference of the circle will be 56.52 m.
And, The area of a circle will be 254.34 m².
What is Circle?
The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.
Given that;
The radius of circle = 9 m
Now,
Since, The area of circle = πr²
Hence, The area of circle = 3.14 × 9²
= 3.14 × 81
= 254.34 m²
And, The circumference of the circle = 2πr
= 2 × 3.14 × 9
= 56.52 m
Thus, The circumference of the circle will be 56.52 m.
And, The area of a circle will be 254.34 m².
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which inequality is less than the quadratic function with zero -7 and 1 and includes the point (2,27/2) on the boundary?
y<-3/2 x²+9 x+21/2
y<3/2 x²+9 x-21/2
y<-3/2 x²-9 x+21/2
y<3/2 x²-9 x-21/2
The inequality is less than the quadratic function with zero -7 and 1 and includes the point (2,27/2) quadratic function is
y < 3/2 x² + 9 x - 21/2How to show the inequality requiredThe problem seeks the inequality that will be less than the quadratic function with zero -7 and 1 and includes the point (2, 27/2).
Therefore the quadratic function should have zeros at -7 and 1
Substituting the point (2, 27/2) to the quadratic function to check the one that gives the points shows that
3/2 x² + 9 x - 21/2 at x = 2
3/2 * 2² + 9 * 2 - 21/2
= 27/2
Solving for the zeros in the quadratic function 3/2 x² + 9 x - 21/2
shows that the zero are at
x = -7 and
Therefore the inequality that meets the condition is y < 3/2 x² + 9 x - 21/2
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TIME SENSITIVE
(2,6)
x-y=4
(a) write an equation of the line that passes through the point and is parallel to the given line
(b) write an equation of the line that passes through the point and is perpendicular to the given line
Answer:
(a) Equation of parallel line is y = x + 4
(b) Equation of perpendicular line is y = - x + 8
Step-by-step explanation:
Given line is
x - y = 4
Rewrite this as
x - 4 = y
or
y = x - 4
This is the equation of a line in slope intercept form which is generally
y = mx + b
where m is the slope and b the y-intercept
Here m = 1, b = -4
(a) A parallel line will have the same slope but a different y-intercept
So for starters we can write the equation of the parallel line as
y = x + b
Since this passes through (2,6) sub 2 for x and 6 for y to get
6 = 2 + b
==> b = 4
Equation of the line is
y = x + 4
(b) A perpendicular line will have as slope -1/m where m is the slope of the original line
Here m = 1, so -1/m = -1/1 = -1
Equation will be of the form
y = -x + b
Since this also passes through (2,6)
6 = -2 + b
==> b = 6 + 2 = 8
Equation of perpendicular line is
y = -x + 8
In a recent poll, 280 people were asked if they liked dogs, and 42% said they did. Find the margin of error of this poll, at the 90% confidence level.
If in a recent poll, 280 people were asked if they liked dogs, and 42% said they did. the margin of error of this poll, at the 90% confidence level is 0.05.
How to find the margin of error?Sample proportion =42%(phat)
Sample size =280
Standard error = √(0.42×( 1- 0.42)/280)
Standard error = √(0.42×(.58)/280)
Standard error =0.029
Hence,
Margin of error = z-critical for 90% confidence level× Standard error
Zα/2 = 1.645 for 90% confidence
Margin of error = 1.645 × 0.029
Margin of error = 0.047
Margin of error =0.05 (Approximately)
Therefore the margin of error is 0.05.
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a line with a slope of -3 passes through the point -3,8 what is its equation in slope intercept form
what are the cooridinates of the point that is 1/6 of the way from A(14, -1) to B(-4, 23)?
Answer:
(11, 3)
Step-by-step explanation:
You want the point 1/6 of the way from A(14, -1) to B(-4, 23).
Dividing pointThe point P that divides AB into the fractional parts k and (1 -k) is ...
P = (1-k)A +kB
ApplicationFor k = 1/6, the dividing point is ...
P = (5/6)(14, -1) +(1/6)(-4, 23) = (5·14 -4, 5(-1) +23)/6 = (66, 18)/6
P = (11, 3)
The point 1/6 of the way from A to B is (11, 3).
Sam and Liam raced up and then back down a hill. Sam's average speed up the hill
was 1 mph, and his average speed down the hill was 9 mph. Liam ran up the hill
and down the hill at 2 mph. If the trail from the bottom to the top of the hill is 1
mile long, how much time did it take each boy to finish? Who was the winner?
Using the speed formula, we know that Liam finished in 1 hour as opposed to Sam's 1/ 19 hours, earning him the victory.
What is speed?Speed is calculated as follows: speed = distance * time.
Knowing the units for distance and time is necessary to calculate the units for speed.
The units will be in meters per second (m/s) in this example since the distance is measured in meters (m) and the time is measured in seconds (s).
So, the speed formula is:
Speed = Distance/Time
Taking time for the subject:
Time = distance/Speed
Now, Sam:
Time spent ascending the hill = 1/1 = 1 hour
Time spent descending the hill = 1/9 hours
Sam spent 1 1/the9 total minutes climbing and descending the hill.
Now, Lian:
Time spent ascending the hill = 1/2 hours
Time spent descending the hill = 1/2 hour
Total time spent ascending and descending the hill = 1/2 + 1/2 = 1 hour
Sam, therefore, took 1/9 hour longer than Liam.
Therefore, using the speed formula, we know that Liam finished in 1 hour as opposed to Sam's 1/ 19 hours, earning him victory.
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