The amount deposited is 7500 in the bank with 3. 5% simple interest on a savings account.
To solve this problem, we need to use the formula for simple interest:
I = P * r * t
Where:
I = Interest earned
P = Principal (amount deposited)
r = Interest rate per year (as a decimal)
t = Time (in years)
In this case, we know that the interest rate is 3.5% or 0.035 as a decimal.
We also know that the interest earned is $525, and the time period is 2 years.
So, we can plug in these values and solve for the principal:
525 = P * 0.035 * 2
Simplifying the equation, we get:
525 = 0.07P
Dividing both sides by 0.07, we get:
P = 7500
Therefore, the amount deposited in the savings account was $7500.
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Determine whether each set of measures can be the measures of the sides of a
right triangle. then state whether they form a pythagorean triple.
13. 12, 16, 20
14. 16, 30, 32
15. 14, 48, 50
16.
2 4 6
5' 5' 5
17.2v6,5,7
18. 2v2, 2v7,6
13. They can, in fact, be the lengths of a right triangle's sides. A Pythagorean triadic is formed by them.
14. They can, in fact, be the lengths of a right triangle's sides. A Pythagorean triadic is formed by them.
15. They can, in fact, be the lengths of a right triangle's sides. A Pythagorean triadic is formed by them.
16.They're unfit to be a right triangle's side measures. The forecourt of the longest side( 62 = 36) can not be calculated by adding the places of the two lower sides( 22 42 = 20).
17. They can, in fact, be the lengths of a right triangle's sides. A Pythagorean triadic is formed by them.
18. They're unfit to be a right triangle's side measures.
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The rate of change dP/dt of the number of students who heard a rumor is modeled by a logistic differential equation. The maximum capacity of the school is 732 students. At 12 PM, the number of students who heard the rumor is 227 and is increasing at a rate of 24 students per hour. Write a differential equation to describe the situation. dP/dt =?
The differential equation to describe the situation is:
dP/dt = 0.000508 * P * (1 - P/732)
We can write a logistic differential equation to describe the rate of change of students who heard the rumor. The equation is:
dP/dt = k * P * (1 - P/M)
where dP/dt is the rate of change in the number of students who heard the rumor, k is a constant, P is the number of students who have heard the rumor at a given time, and M is the maximum capacity of the school (732 students).
At 12 PM, P = 227 and dP/dt = 24 students per hour. We can plug these values into the equation:
24 = k * 227 * (1 - 227/732)
Now, solve for k:
k ≈ 0.000508
So, the differential equation to describe the situation is:
dP/dt = 0.000508 * P * (1 - P/732)
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The distance from Atlanta, Georgia, to Boise, Idaho is 2,214 miles. The distance from Atlanta, Georgia, to Houston, Texas is 789 miles. How much farther is it from Atlanta to Boise than from Atlanta to Houston?
Answer:
1,425 miles
Step-by-step explanation:
To find out how much farther it is from Atlanta to Boise than from Atlanta to Houston, we need to subtract the distance from Atlanta to Houston from the distance from Atlanta to Boise:
[tex]\sf:\implies 2,214\: miles - 789\: miles = \boxed{\bold{\:\:1,425\: miles\:\:}}\:\:\:\green{\checkmark}[/tex]
Therefore, it is 1,425 miles farther from Atlanta to Boise than from Atlanta to Houston.
KAP
1 IN PU
1. Luis parents give him x dollars for his monthly allowance. Each month, he must
pay $35 for his cell phone. One-sixth of the remaining money can be spent on
entertainment. Which function can be used to find the amount in dollars Luis can
spend on entertainment?
A f(x) =*735
B. F(x) = 35x - 3
C. F(x) = 35 - $
D. F(x) = § - 35
The function can be used to find the amount in dollars Luis can spend on entertainment is 1/6(x-35)
Total amount of money luis get from his parent = x dollars
The amount luis has to pay to his parents for his cell phone is $35
After giving money for cell phone the amount of money left with luis will be x - 35
so, remaining money with luis = x - 35
One sixth of the remaining money which can be spent on entertainment by luis is 1/6(x-35)
The function can be written in the form of
f(x) = 1/6(x-35)
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fast pls
in When calculating 4-√x+15 what option you get x?-1 lim X>1 1 the process -1 A) lim (x + 1) (4 + V1 +15) -1 B) lim 21 (1+1) (4 - V1-15) C) lim 16 - 2 - 1)(4+r+15) D) lím 16-1 (12 - 1) (4 - Vr+15)
None of the options are correct, and the value of x is simply 1.
Start with the given expression: 4 - √x + 15
Substitute x with the limit value of 1: 4 - √1 + 15 = 18
Therefore, the limit of the given expression as x approaches 1 from the right is 18.
To verify this result using the provided options, we can simplify each option and check which one equals 18 as x approaches 1 from the right.
Option A simplifies to (2/√2) + 2, which equals √2 + 2, not equal to 18.
Option B simplifies to 2(4 - √15), which equals 2(4 - 3.87), approximately equal to 2.27, not equal to 18.
Option C simplifies to 3(4 + √15), which equals 3(4 + 3.87), approximately equal to 23.61, not equal to 18.
Option D simplifies to 3(3)(3), which equals 27, not equal to 18.
Therefore, none of the options are correct, and the value of x is simply 1.
none of the options are correct, and the value of x is simply 1.
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16. Justin is joining a gym. The gym is currently offering a discount on the fee to join and on the monthly rate.
The discounted price,in dollars,the gym charges can be represented by the equation y=10x+5
a. What are the slope and the Y-intercept of the equation? What do the slope and the Y-intercept each represent in this equation?
Answer:
The equation y = 10x + 5 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
The equation y = 10x + 5 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.Slope (m) = 10
The equation y = 10x + 5 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.Slope (m) = 10Y-intercept (b) = 5
The equation y = 10x + 5 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.Slope (m) = 10Y-intercept (b) = 5The slope represents the rate of change of the monthly rate with respect to the fee to join. For every increase of $1 in the fee to join, the monthly rate increases by $10.
The equation y = 10x + 5 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.Slope (m) = 10Y-intercept (b) = 5The slope represents the rate of change of the monthly rate with respect to the fee to join. For every increase of $1 in the fee to join, the monthly rate increases by $10.The y-intercept represents the initial cost of joining the gym. It is the amount that the gym charges even if the fee to join is $0. In this case, the gym charges $5 to join.
A student usally saves $20 a month. He would like to reach a goal of saving $350 in 12 months the students writes the equation 350=12(x + 20) to represent this situation
Answer: x=55/6 or 55 over 6
Step-by-step explanation: Step 1: Distribute:
- 350= 12(x+20)
- 350= 12x + 240
Step 2: Subtract 240 from both sides:
- 350-240= 12x+240-240
Step 3: Simplify:
Subtract the numbers: 350-240= 12x+240-240= 110=12x+240-240
Subtract again: 110=12x+240-240= 110=12x
Step 4: Divide both sides by the same factor:
110=12x= 110/12= 12x/12
Step 5: Simplify:
- Divide the numbers: 110/12=12x/12= x=55/6=12x/12
- Cancel terms that are in both the numerator and denominator: 55/6=12x/12= 55/6=x
- Move the variables to the left: 55/6=x = x=55/6
Answer: x=55/6 or 55 over 6
The profit ( in hundreds of dollars) from selling units of a product is given by
the profit function P(x) = x^2/ 2x + 1 Find the marginal profit when 4 units are produced
and sold and interpret your answer using words, numbers and units. Be very specific.
(Round the number part of your answer to the nearest cent.)
the marginal profit when 4 units are produced and sold is approximately -$69.14. This means that when producing and selling the 4th unit, the profit will decrease by $69.14.
To find the marginal profit when 4 units are produced and sold, we first need to find the derivative of the profit function P(x) with respect to x. The given profit function is:
P(x) = (x^2) / (2x + 1)
Now, let's find its derivative, which represents the marginal profit function:
dP/dx = (d/dx(x^2))/(2x + 1) - (x^2)(d/dx(2x + 1))/((2x + 1)^2)
First, find the derivative of x^2 and 2x + 1:
d/dx(x^2) = 2x
d/dx(2x + 1) = 2
Now, substitute these values into the marginal profit function:
dP/dx = (2x)/(2x + 1) - (x^2)(2)/((2x + 1)^2)
Next, we'll find the marginal profit when 4 units are produced and sold. So, let's substitute x = 4 into the marginal profit function:
dP/dx(4) = (2 * 4)/(2 * 4 + 1) - (4^2)(2)/((2 * 4 + 1)^2)
dP/dx(4) = (8)/(9) - (32)(2)/(81)
dP/dx(4) = (8 - 64)/(81)
dP/dx(4) = -56/81
Since the profit is in hundreds of dollars, we need to multiply the marginal profit by 100 to get the value in dollars. Then, round to the nearest cent:
Marginal profit in dollars = (-56/81) * 100 ≈ -$69.14
So, the marginal profit when 4 units are produced and sold is approximately -$69.14. This means that when producing and selling the 4th unit, the profit will decrease by $69.14.
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The prism shown has a surface area of 1,500 mm squared. What is the height, h, of the prism?
The height of the triangular base prism is 10 mm.
How to find the surface area of a prism?The prism above is a triangular base prism. The surface area of the prism can be calculated as follows:
surface area of the triangular prism = (a + b + c)l + bh
where
a, b, and c are the side of the triangular basel = height of the prismb = base of the triangleh = height of the triangleTherefore,
surface area of the triangular prism = (20 + 30 + 40) + 20 × 30
1500 = 90l + 600
1500 - 600 = 90l
90l = 900
divide both sides by 90
l = 900 / 90
l = 10 mm
Therefore,
height of the prism = 10 mm
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A tank in the shape of a hemisphere has a diameter of 8 feet. If the liquid that fills the tank has a density of 86 pounds per cubic foot, what is the total weight of the liquid in the tank, to the nearest full pound?
Answer:
209.07 pounds
Step-by-step explanation:
radius= 8÷2=4 feet
volume of hemisphere=((4/3)×(22/7)×r^3)/2
=134.09 cubic feet
Mass=density × volume
=86×134.09
=209.07 pounds
1)Change point A in the scatterplot to point (1,12). Calculate the correlation coefficient and note how much it differs from .96. (2)Change point A back to (1,2) and change point B to (4,15). Calculate the correlation coefficient and note how much it differs from .96. Did the correlation coefficient change more when the point you raised 10 units was in the middle of the scatterplot or at the edge of the scatterplot? Why do you think this is so? (3)Move only one point and make the correlation coefficient become negative. Write about what you did and why it made the correlation go negative.(4) Suppose you had a scatterplot with only two points. Assuming your two points don't define either a horizontal line (both y-values the same) or a vertical line (both x-values the same), what is the correlation coefficient? Why do you think this is true? What happens as you try different points (again, without defining a horizontal or vertical line)?(5)Enter the points (1,2) and (3,2) — this defines a horizontal line. Try to calculate the correlation coefficient. What did your graphing calculator tell you? What happened?(6) Enter the points (1,2) and (1,3) — this defines a vertical line. Try to calculate the correlation coefficient. What did your graphing calculator tell you? What happened? The following scatterplot was constructed by reversing the x- and y-values in the original scatterplot. Without calculating the new correlation coefficient, what do you think r is? Why? (7)Graph depicts 16 scatter plots on a coordinate plane without coordinate points. 7 scatter plots in quadrant 3, 1 scatter plot in quadrant 4, and 8 scatter plots in quadrant 1. The following scatterplot was constructed by taking the negative of each x-value in the original scatterplot. Without calculating the new correlation coefficient, what do you think r is? Why? What would the correlation coefficient be if we took the negative of all the x-values and all the y-values? Graph depicts 15 scatter plots on a coordinate plane without coordinate points. 7 scatter plots in quadra
The new regression coefficient is about 0.663, viz. lesser than the previous regression coefficient by 0.297. Thus, a single outlier creates a significant drop in the correlation
How to solveChanging A to (1,12) gives below scatterplot and regression parameters
(check image)
2. In this case, r is about 0.766, a drop of 0.194 which is substantial, but lower than the previous drop. The regression coefficient changed much more when the outlier was in the middle of the scatterplot. This happens because the data series itself is increasing.
So the effect of 10 points in a middle point is much more of an outlier compared to when this 10-point increase happens for the highest value of x. Hence, the r value drops more in the former case.
3. r can become negative if drop the point B to a highly negative y-value. Consider taking it to (4, -50). Then we get the following regression parameters
We obtain r = -0.275. Since the expected y-value was highest for point B, so changing it drastically to a large negative value leads to a negative correlation between the two variables.
4. With only two points that are parallel to neither of the axes, the correlation coefficient is always exactly either 1 or -1. The correlation is 1 if the slope of the line joining the two points is positive, and -1 if the slope is negative.
That is, there is always either a perfect positive correlation or a perfect negative correlation. This is so because there is always a unique line joining two points, which leads to a perfect correlation between them. Even by differing the pairs, this relation shall always hold true.
5. If the points are parallel to the X axis, we should obtain r=0, because it indicates no relation between the variables. So points (1, 2) and (3, 2) lead to r=0. This can be verified using any calculator.
A vertical line also leads to r=0. Since the y value does not change, so no correlation can be established. Actually, it is just like flipping the x and y variables, and we know flipping does not change the correlation coefficient. So we should obtain r=0 even for a vertical line.
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Hallar la altura de una asta bandera, si un estudiante la observa desde un punto a, con un ángulo de 30° y entre el estudiante y la asta hay una distancia de 10m.
Answer:
The height of the flagpole is approximately 5.774 meters.
Step-by-step explanation:
Let's call the height of the flagpole h. We can use trigonometry to set up the following equation:
tan(30°) = h/10
Simplifying this equation, we get:
h = 10 tan(30°)
Using a calculator, we find that tan(30°) ≈ 0.5774, so:
h ≈ 5.774 meters
Therefore, the height of the flagpole is approximately 5.774 meters.
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Shandra has $760 to spend at a bicycle store for some new gear and biking outfits. Assume all prices listed include tax. ⢠She buys a new bicycle for $433. 54. ⢠She buys 2 bicycle reflectors for $18. 41 each and a pair of bike gloves for $10. 76. ⢠She plans to spend some or all of the money she has left to buy new biking outfits for $66. 40 each. Write and solve an inequality which can be used to determine o, the number of outfits Shandra can purchase while staying within her budget. â
We can use an inequality to determine the number of biking outfits Shandra can purchase while staying within her budget.
Let o represent the number of outfits she can purchase. Here are the given terms and costs:
- Initial budget: $760
- Bicycle cost: $433.54
- 2 reflectors cost: 2 * $18.41 = $36.82
- Bike gloves cost: $10.76
- Outfit cost: $66.40 each
Now, we can set up the inequality:
760 >= 433.54 + 36.82 + 10.76 + 66.40 * o
First, combine the constants:
760 >= 481.12 + 66.40 * o
Now, subtract 481.12 from both sides:
278.88 >= 66.40 * o
Finally, divide both sides by 66.40:
o <= 4.2
Since Shandra can only purchase whole outfits, the maximum number of outfits she can buy is 4. So the inequality representing this situation is o <= 4.
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The angle of depression from the top of a 150m high cliff to a boat at sea is 7°. How much closer to the cliff must the boat move for the angle of depression to become 19°?
The boat must move 785.82 m closer to the cliff for the angle of depression to become 19°.
We need to find how much closer to the cliff the boat must move for the angle of depression to change from 7° to 19°.
Calculate the distance from the boat to the base of the cliff at 7° angle of depression.
Using the tangent function, we have:
tan(angle) = height/distance
tan(7°) = 150m/distance
distance = 150m/tan(7°)
distance=1221.49
Calculate the distance from the boat to the base of the cliff at 19° angle of depression.
Using the tangent function, we have:
tan(angle) = height/distance
tan(19°) = 150m/distance
distance = 150m/tan(19°)
distance=435.6665
Calculate the difference between the two distances to find out how much closer the boat must move.
difference = distance at 7° angle of depression - distance at 19° angle of depression
Plugging in the values from Steps 1 and 2, we get:
difference = (150m/tan(7°)) - (150m/tan(19°))
difference=1221.49-435.6665
difference=785.8235
After calculating, we find that the boat must move approximately 785.82 meters closer to the cliff for the angle of depression to change from 7° to 19°.
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callum says 300cm2 is the same as 3m2 because there are 100cm in 1m so you divide by 100 callums method is wrong explain why
Answer:
Callum’s method is incorrect because he is confusing the conversion of linear units with the conversion of square units. There are indeed 100 cm in 1 m, but when converting square units, you need to square the conversion factor. So 1 m² is equal to (100 cm)² or 10,000 cm². Therefore, 300 cm² is equal to 0.03 m², not 3 m².
Step-by-step explanation:
Answer:
The answer is wrong because the 3m is still squared if you divided it by 100 then it should only be 3m not 3m^2.
Step-by-step explanation:
I am not 100% sure this is correct so please dont get mad at me.
PLEASE HELP (40 POINTS)
The coordinates of the points N and L are N = (-2d, 0) and L = (-4f, g)
Calculating the coordinates of N and LFrom the question, we have the following parameters that can be used in our computation:
M = (-2d - 4f, g)
O = (0, 0)
ON = 2d
Given that
ON = 2d
Then it means that
N = (-2d, 0)
For the point L, we have
LO = MN
Where
LO = √[(x - 0)² + (y - 0)²] i.e. the distance formula
LO = √[x² + y²]
Next, we have
MN = √[(-2d - 4f + 2d)² + (g - 0)²] i.e. the distance formula
MN = √[(-4f)² + g²]
So, we have
LO = MN
√[x² + y²] = √[(-4f)² + g²]
By comparison, we have
x = -4f and y = g
This means that the coordinates of point L = (-4f, g)
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What's 2x = 20 if 4y = 80
Answer:
x=10 and y=20, 2x=20=y
Step-by-step explanation:
2x=20 | Divide by 2 on both sides
x=10
4y=80 | Divide by 4 on both sides
y=20
Answer: 40
80 ÷ 2 = 20, so x = 40.
To check your answer;
40 x 2 = 80
For each pair of numbers, decide if lines with these gradients are perpendicular or not. a) 5 and 1/ 5 b) 2/3 and -1/3 c) and -1/1 d) - and 3
The pair of numbers 3/5 and -5/3, -1/3 and 3 are perpendicular because two lines are perpendicular if and only if the product of their gradients is -1.
Each pair of numbers, we have to decide if lines with these gradients are perpendicular or not
Two lines are perpendicular if and only if the product of their gradients is -1.
For 5 and 1/5
The product is 1 which is not -1, so these are not perpendicular.
For 3/5 and -5/3
The product is -1 so these are perpendicular
For 1/4 and -1/4
The product is -1/16 so these are not perpendicular.
For -1/3 and 3
The product is -1 so these are perpendicular
Hence, the pair of numbers 3/5 and -5/3, -1/3 and 3 are perpendicular.
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ALGEBRA
Farha Gadhia has applied for a $100,000 mortgage loan
at an annual interest rate of 6%. The loan is for a period of 30 years
and will be paid in equal monthly payments that include interest.
Use the monthly payment formula to find the payment.
Penny decided to travel to Palawan. The airplane flew at an average rate of 300 miles per hour and covered 1500 miles. How long will the flight will take? *
A. 3 hours
B. 4 hours
C. 5 hours
D. 6 hours
The time it will take the flight is C) 5 hours.
To solve this problem, we can use the formula: distance = rate x time. In this case, we know that the distance is 1500 miles and the rate (or speed) is 300 miles per hour. We can rearrange the formula to solve for time: time = distance / rate. Plugging in the values we have, we get:
time = 1500 miles / 300 miles per hour
time = 5 hours
Therefore, the correct answer is C. It will take Penny 5 hours to fly from her starting point to Palawan at an average speed of 300 miles per hour. This calculation assumes that the plane maintains a constant speed throughout the entire flight, which may not be the case due to factors such as wind and turbulence.
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Mai drew the design shown below. Each
rectangle in the design has the same
area. Each rectangle is what fraction of
the area of the complete design?
Each rectangle is 1/3 of the area of the complete design.
What fraction of the area of the complete design?A fraction represents the parts of a whole or collection of objects e.g. 3/4 shows that out of 4 equal parts, we are referring to 3 parts.
Looking at the design, you will be notice that the main (bigger) rectangle is divided to three smaller rectangles. Thus, each rectangle is one out of three rectangles i.e. 1/3.
Therefore, each rectangle is 1/3 of the area of the complete design.
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Complete Question
Check attached image
(2) 44% of the students in Prof. Young's class are Liberal Arts major, 64% major in Business Administration, and 39% major in both. Compute the probability that a student selected at random in Prof. Young's class major in Liberal Arts or Business Administration.
The probability that a student selected at random from the class majors in Liberal Arts or Business Administration is 0.69 or 69%.
Let us say that the event that a student is a Liberal Arts major is 'LA' and the event that a student is a Business Administration major is 'BA'.
Now according to the question, we know that 44% of students are Liberal Arts majors, 64% of students are Business Administration majors,
and 39% of students are majoring in both.
With this information, we can say that:
P(LA) = 44% = 0.44
P(BA) = 64% = 0.64
• P(LAN BA) = 39% = 0.39
Now in order to find the probability that a student
is selected at random majors in either Liberal Arts
or Business Administration, we need to compute the value of P(LA U BA), which is the probability of either of the event happening.
The formula for the probability of the union of two events can be used to find the union that is: PILA U BA) = P(LA)+P(BA)-PILAN BA)
Now by substituting the values in the above equation, we get:
P(LA U BA) = 0.44 + 0.64 -0.39
P(LA U BA) = 0.69
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The area of the triangle below is \frac{2}{25}
25
2
square feet. What is the length of the base? Express your answer as a fraction in simplest form.
1/5 f
The length of the base of the given triangle can be simplified as 2√2/5 feet, which is equivalent to √8/5 feet.
What is the length of the base of a triangle if its area is (2/25) * 252 square feet and the height is twice the length of the base?We are given that the area of the triangle is (2/25) * 252 square feet.
Let the length of the base be x. Then, the height of the triangle can be expressed as (2/5)x, since the base divides the triangle into two equal parts.
The area of the triangle is given by the formula A = (1/2)bh, where b is the length of the base and h is the height of the triangle.
Substituting the given values, we get:
(1/2)x(2/5)x = (2/25)*252
Simplifying this equation, we get:
(1/5)x²= 20.16
Multiplying both sides by 5, we get:
x² = 100.8
Taking the square root of both sides, we get:
x =√(100.8)
Simplifying this expression, we get:
x = √(25*4.032)x = 5*√(4.032)x = (5/5)*√(4.032)x = 1*√(4.032)Therefore, the length of the base is √(4.032) feet, which can be expressed as a fraction in simplest form as 2√(2)/5 feet.
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I need help with this question.
Answer: The answer is (C
Step-by-step explanation:
Bob and two friends each were able to juggle with bean bags for 3/4 of a minute. How long did they juggle together? No decimals pls!
Answer:
Step-by-step explanation:
They each juggled for 3/4 of a minute
There were 3 people in total
3 people times 3/4 of a minute equals 2 1/4 minutes
Math geometry question please help
Answer:
LP is a raym∠LMO = 90°m∠LPO = 90°m∠MLP = 26°arc MP = 154°arc MNP = 206°Step-by-step explanation:
You want various angle and arc measures in the given figure.
RelationshipsThe relevant angle relationships are ...
a radius to a point of tangency makes a right angle with the tangentan arc has the same measure as its central anglethe sum of the arcs of a circle is 360°the sum of angles in a quadrilateral is 360°an angle is formed from two rays whose endpoints are the vertex of the angleThese answer the given questions as follows:
LP is a ray.
The angle at M is 90°.
The angle at P is 90°.
The angle at L is 360° -90° -90° -154° = 26°
Arc MP has the same measure as angle MOP, 154°
Arc MNP completes the circle, so is 360° -154° = 206°
Please help with this question. I am offering 40 points. You can use the word box above to answer the questions.
We define [tex]9^{\frac{1}{2} }[/tex] to be the square root of nine. This means that [tex](9^{\frac{1}{2} })^{2}[/tex] must be equal to nine.
By using the properties of exponents to explain why [tex]8^{\frac{1}{3} }=2[/tex];
Statement Reasons_____________________
[tex]8^{\frac{1}{3} }=2[/tex] Given
[tex](8^{\frac{1}{3} })^3=2^3[/tex] Exponent property of equality.
[tex]8^{\frac{1}{3} \times 3}=2^3[/tex] Exponent property of a power.
8 = 8 simplify
What is an exponent?In Mathematics, an exponent refers to a mathematical operation that is typically used in conjunction with an algebraic expression in order to raise a quantity to the power of another.
This ultimately implies that, an exponent is represented by the following mathematical expression;
bⁿ
Where:
the variables b and n are numerical values (numbers) or an algebraic expression.
n is referred to as a superscript or power.
By applying the multiplication and division law of exponents for powers to each of the expressions, we have the following:
[tex]9^{\frac{1}{2} }[/tex] = √9 (square root of nine)
[tex](9^{\frac{1}{2} })^{2}=(9^{\frac{1}{2} \times 2})=9^1 = 9[/tex]
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Aleks and Melanie used a protractor to measure the angle below. Aleks thinks the angle measures 50° but Melanie says it is actually 130°. Their teacher confirms that Melanie has the correct answer. What mistake did Aleks make while measuring the angle?
The mistake, Aleks made, while measuring the angle is, he measure the angle from the wrong side of the line.
Angle is a dimensionless vector quantity, that is, it is very important, to take care of the directions, while measuring the angle.
That is, to measure the angle, say ∠ABC, the 0°(reference) line of the protractor, must be on one either AB or BC, to measure the angle rightly.
And since the angles between two lines are supplementary in nature, that is, the two angles will add up to make 180°, that is why, the angle measure by Alek and Melanie, add up to make 180°.
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Need this fast
A) -24 Solve lim 69²-24 B) 4 a+2 2-a C) 24 D) - 4
The limit of (69²-24) as x approaches infinity is equal to infinity.
As x approaches infinity, the value of (69²-24) becomes very large, and it goes to infinity. Therefore, the limit of (69²-24) as x approaches infinity is infinity.
B) The limit of (4a+2)/(2-a) as a approaches 2 from the left is equal to -6 and as a approaches 2 from the right is equal to 6.
As a approaches 2 from the left, the denominator (2-a) approaches zero from the negative side, and the numerator (4a+2) approaches -6. Therefore, the limit of (4a+2)/(2-a) as a approaches 2 from the left is -6.
As a approaches 2 from the right, the denominator (2-a) approaches zero from the positive side, and the numerator (4a+2) approaches 6. Therefore, the limit of (4a+2)/(2-a) as a approaches 2 from the right is 6.
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find the general solution of the following linear system. y′ = [2 2 −4 2 −1 −2 4 2 −6] y with λ = −1,−2,−2
To find the general solution of the linear system y' = [2 2 -4; 2 -1 -2; 4 2 -6] y, we need to first find the eigenvectors and eigenvalues of the coefficient matrix A = [2 2 -4; 2 -1 -2; 4 2 -6].
Using the characteristic equation, we can find the eigenvalues:
det(A - λI) = 0
=> det([2-λ 2 -4; 2 -1-λ -2; 4 2 -6-λ]) = 0
=> (2-λ)[(-1-λ)(-6-λ) - 4] - 2[(-2)(-6-λ) - 8] + 4[2(-1-λ) - 4] = 0
=> λ^3 - + 8λ - 4 = 0
=> (λ-1)(λ-2[tex])^2[/tex] = 0
Thus, λ = 1, 2 (with multiplicity 2). For each eigenvalue, we need to find a corresponding eigenvector.
For λ = 1, we need to find the null space of the matrix (A - λI):
A - λI = [1 2 -4; 2 -2 -2; 4 2 -7]
=> R2 <- R2 - 2R1, R3 <- R3 - 4R1
[1 2 -4; 0 -6 6; 0 -6 9]
=> R3 <- R3 - R2
[1 2 -4; 0 -6 6; 0 0 3]
So, we have a basic eigenvector of the form [4,-2,1]^T. To obtain a linearly independent eigenvector, we use the method of generalized eigenvectors. We need to find a vector v such that (A - λI) v = u, where u is the basic eigenvector.
(A - λI) v = u
=> [1 2 -4; 2 -2 -2; 4 2 -7] v = [4; -2; 1]
=> R2 <- R2 - 2R1, R3 <- R3 - 4R1
[1 2 -4; 0 -6 6; 0 -6 9] v = [4; -2; 1]
=> R3 <- R3 - R2
[1 2 -4; 0 -6 6; 0 0 3] v = [4; -2; 1]
=> -6v2 + 6v3 = -2
=> 3v3 = 1
=> 2v2 - 4v3 = -2
=> v2 = 0
So, we have v = [0; 1/3; 2/3[tex]]^T[/tex] as the second eigenvector corresponding to λ = 1.
For λ = 2, we need to find the null space of the matrix (A - λI):
A - λI = [0 2 -4; 2 -3 -2; 4 2 -8]
=> R1 <-> R2
[2 -3 -2; 0 2 -4; 4 2 -8]
=> R3 <- R3 - 2R1
[2 -3 -2; 0 2 -4; 0 8 -12]
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