Answer:
The coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].
Step-by-step explanation:
It is given that Hillman Peak, Garfield Peak, and Cloudcap are three mountain peaks on the rim of the lake. The peaks are located in a coordinate plane at H(-4,1), G(-2,-3), and C(5,-2).
If we joint these points, then we get a triangle and the center of a triangle is known as centroid.
The formula for centroid of a triangle is
[tex]Centroid=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})[/tex]
The centroid of the triangle H(-4,1), G(-2,-3), and C(5,-2) is
[tex]Centroid=(\frac{-4-2+5}{3},\frac{1-3-2}{3})[/tex]
[tex]Centroid=(\frac{-1}{3},\frac{-4}{3})[/tex]
Therefore the coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].
Write an equation and solve: Twice a number, increased by 3 is 7.
Write a single algebraic rule for the series of transformations: a reflection about the x-axis, a rotation of 90 degrees clockwise, and a translation of 4 units right and 2 units down.
Find the first six terms of the sequence. a1 = -3, an = 2 ● an-1
The height of a tree in feet over x years is modeled by the function f(x). f(x)=301+29e−0.5x which statements are true about the growth of the tree? select each correct answer. the tree's maximum height is limited to 30 ft. the tree is initially 2 ft tall. between the 5th and 7th years, the tree grows approximately 7 ft. after growing 15 ft, the tree's rate of growth decreases.
Final answer:
The tree's maximum height is not limited to 30 ft but approaches 301 ft. Initially, the tree is 30 ft tall, not 2 ft. To verify the growth between the 5th and 7th years, one must calculate f(5) and f(7). The rate of growth indeed decreases over time.
Explanation:
When analyzing the function f(x) = 301 + 29e^{-0.5x} to understand the growth pattern of a tree, it is apparent that some statements about the tree's growth can be identified as true or false.
The tree's maximum height is not limited to 30 ft; instead, the model suggests that the height approaches 301 ft as x increases indefinitely, because the exponential term decreases towards zero.The tree is initially 30 ft tall, not 2 ft tall, since f(0) = 301 + 29 × 1 = 330 ft.Between the 5th and 7th years, the growth can be calculated using f(5) and f(7) to determine if it grows approximately 7 ft during that period.After growing 15 ft, the rate of growth would decrease since the exponential function's growth rate decreases as x increases.Which box plot represents the data?
135, 149, 156, 112, 134, 141, 154, 116, 134, 156
The box plot that represents the data is option B.
What is a box plot?A box plot is known to be a method that is employed to demonstrate the locality and skewness of numerical data. This is carried out graphically and done through the quartiles of the given numerical data.
Rearranging the given data, we have:
112,116, 134, 134, 135, 141, 149, 154, 156, 156
The minimum value after rearranging = 112.
First Quartile, Q1 = 134.
Second Quartile, Q2 or Median = 135+141/2 = 276/2 = 138
Third Quartile, Q3 = 154
Largest figure = 156
The answer is the second figure.
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24 inches is 250% of what length
Which transformation is a rotation?
Answer:
the answer is c.
Step-by-step explanation:
If you shift the linear parent function, f(x)=x, down 7 units what is the equation of the new function
f(x)=x-7 is indeed the correct answer.just got it right
i need help!!!!!!!!!!!!!!?!!!
Rewrite 0.85 as a fraction in lowest terms.
Simplify i to the power 41
Answer:
It should be just "i"
What is the solution set of (x - 2)(x - 3) = 2?
a. {1, 4}
b. {2, 3}
c. {4, 5}
students have 40 minuets to complete 50 questions on the aspire ___ test
A) math
B) reading
C) english
D) writing
Answer:
english C)
Step-by-step explanation:
sabendo que x2 + y2=74 e que xy=35, calcule valor de (x - y2)2
Find the value of the lesser root of x^2-6x+8=0
Answer:
The value of lesser root is:
x=2
Step-by-step explanation:
We are given a quadratic equation in terms of variable " x " as:
[tex]x^2-6x+8=0[/tex]
We know that for any quadratic equation of the type:
[tex]ax^2+bx+c=0[/tex]
The roots of x are calculated as:
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Here we have:
a=1 , b=-6 and c=8
Hence, on solving for roots:
[tex]x=\dfrac{-(-6)\pm \sqrt{(-6)^2-4\times 1\times 8}}{2\times 1}\\\\\\x=\dfrac{6\pm \sqrt{36-32}}{2}\\\\\\x=\dfrac{6\pm \sqrt{4}}{2}\\\\\\x=\dfrac{6\pm 2}{2}[/tex]
Hence, we have:
[tex]x=\dfrac{6+2}{2}\ or\ x=\dfrac{6-2}{2}\\\\x=\dfrac{8}{2}\ or\ x=\dfrac{4}{2}\\\\\\x=4\ or\ x=2[/tex]
Hence, the value of lesser root is:
x=2
Factor this trinomial completely.
10x2 + 7x – 12
A. (2x + 3)(5x – 4)
B. (x + 4)(10x – 3)
C. (x + 3)(10x – 4)
D. (2x + 4)(5x – 3)
The trinomial "10[tex]x^2[/tex] + 7x - 12" is factored by finding two numbers that multiply to -120 and add up to 7, which are 15 and -8. Split the middle term, group, and factor by grouping to get (2x + 3)(5x - 4), which is option A.
Explanation:To factor the trinomial "10x2 + 7x − 12" completely, you want to find two binomials that, when multiplied together, will give you the original trinomial. To do this, you will need two numbers that multiply to give you the product of the coefficient of the x2 term (which is 10) and the constant term (which is −12), which equals −120, and also add up to the coefficient of the x term (which is 7).
The two numbers that meet these criteria are 15 and −8, since (15)(−8) = −120 and 15 + (−8) = 7. Next, split the middle term of the trinomial using these two numbers:
10x2 + 15x − 8x − 12
Now, group the terms:
(10x2 + 15x) − (8x + 12)
Factor out the greatest common factor from each group:
5x(2x + 3) − 4(2x + 3)
Finally, factor out the common binomial factor:
(2x + 3)(5x − 4)
Therefore, the correct factorization of the trinomial 10x2 + 7x − 12 is (2x + 3)(5x − 4), which corresponds to option A.
The center of a circle is at (2, -5) and it's radius is 12. What is the equation of the circle?
A.) (x+2)^2 + (y-5)^2 =24
B.) (x-2)^2 + (y+5)^2 = 24
C.) (x+2)^2 + (y-2)^2 = 144
D.) (x-2)^2 + (y+5)^2 = 144
Answer:
The answer is D
Step-by-step explanation:
It is D
Which number would it be if rounded to the nearest foot
7191.51
A. 7191 FT
OR
B. 7192 FT
Please help!!!
Use DeMoivre's theorem to evaluate the expression
[sqrt 3( cos 5pi/3 + i sin 5pi/3)]^4 ? write the answer in rectangular form
a. 9sqrt3/2 + 9/2 i
b. 9sqrt3/2 - 9/2 i
c. -9/2 - 9sqrt3/2 i
d. -9/2 + 9sqrt3/2 i
Answer:
Option d - [tex]\sqrt3(\cos (\frac{5\pi}{3})+i\sin(\frac{5\pi}{3}))^4=-\frac{9}{2}+\frac{9\sqrt3}{2}i[/tex]
Step-by-step explanation:
Given : [tex]\sqrt3(\cos (\frac{5\pi}{3})+i\sin(\frac{5\pi}{3}))^4[/tex]
To find : Use DeMoivre's theorem to evaluate the expression?
Solution :
DeMoivre's theorem state that, for complex number
If [tex]z = r(\cos\theta+ i\sin \theta)[/tex] then [tex]z^n = r^n(\cos n\theta+ i\sin n\theta)[/tex]
We have given,
[tex]\sqrt3(\cos (\frac{5\pi}{3})+i\sin(\frac{5\pi}{3}))^4[/tex]
On comparing [tex]r=\sqrt3[/tex] and n=4
Applying DeMoivre's theorem,
[tex]=(\sqrt3)^4(\cos 4(\frac{5\pi}{3})+i\sin4(\frac{5\pi}{3}))[/tex]
[tex]=9(\cos (\frac{20\pi}{3})+i\sin(\frac{20\pi}{3}))[/tex]
[tex]=9(\cos (6\pi+\frac{2\pi}{3})+i\sin(6\pi+\frac{2\pi}{3}))[/tex]
[tex]=9(\cos (\frac{2\pi}{3})+i\sin(\frac{2\pi}{3}))[/tex]
We know, the value of
[tex]\cos (\frac{2\pi}{3})=-\frac{1}{2},\sin (\frac{2\pi}{3})=\frac{\sqrt3}{2}[/tex]
[tex]=9(-\frac{1}{2}+i\frac{\sqrt3}{2})[/tex]
[tex]=-\frac{9}{2}+i\frac{9\sqrt3}{2}[/tex]
Therefore, Option d is correct.
[tex]\sqrt3(\cos (\frac{5\pi}{3})+i\sin(\frac{5\pi}{3}))^4=-\frac{9}{2}+\frac{9\sqrt3}{2}i[/tex]
Let d(t) =6t^2 be the distance function, find the average velocity from (4, 4.1)
Stan's cookie recipe makes 242424 cookies and calls for exactly 384384384 sprinkles. He is wondering how many sprinkles (p)(p)left parenthesis, p, right parenthesis he will need to make 606060 cookies. He assumes each cookie will have the same number of sprinkles. How many sprinkles does Stan need to make 606060 cookies?
To find the number of sprinkles needed for 60 cookies, determine the number of sprinkles per cookie by dividing 384 by 24, which is 16. Then multiply 16 by 60 to get 960 sprinkles.
The question asks us to calculate the number of sprinkles (p) needed for 60 cookies when a recipe for 24 cookies calls for 384 sprinkles.
To solve this, we start by finding the number of sprinkles per cookie:
Divide the total number of sprinkles (384) by the number of cookies in the recipe (24).Multiply the result by the number of cookies we want to make (60).Therefore:
384 sprinkles ÷ 24 cookies = 16 sprinkles per cookie 16 sprinkles per cookie × 60 cookies = 960 sprinkles for 60 cookiesStan would need 960 sprinkles to make 60 cookies.
Factor each polynomial. Check your answer by distributing. 2x^2+5x+2
The function f(x)=35+15x represents the amount of money, in dollars, mr. lewis earns for working x hours. how much money does mr. lewis earn for working 25 hours?
By substituting x with 25 in the equation f(x)=35+15x, we find out that Mr. Lewis would earn $410 for working 25 hours.
Explanation:The function provided is f(x)=35+15x, which represents the amount of money Mr. Lewis earns for working x hours. In the case of Mr. Lewis working for 25 hours, we substitute the value of x with 25 in the equation to calculate his earnings. Therefore, f(25)=35+15(25)=35+375=$410. Hence, Mr. Lewis would earn $410 for working 25 hours.
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A card is drawn randomly from a standard deck of cards. you win $10 if the card is a spade or an ace. what is the probability that you will win the game?
In circle H, HJ¯¯¯¯¯ is a radius and JK¯¯¯¯¯ is a tangent segment. Which statement must be true?
A) HJ=JK
B) ∠HJK is an obtuse angle.
C) HJ=HK
D) ∠HJK is a right angle.
Cam rode her bike 5 times as far as Dante did. Dante rode 187 meters farther than Michael did. Cam rode 15.25 Kilometers. How many meters did Michael ride? Explain how you know you got your answer
It was found that Michael rode 2863 meters.
To find out how many meters Michael rode, we first need to establish the relationship between the distances Cam, Dante, and Michael rode. According to the problem, Cam rode 5 times as far as Dante did. If Cam rode 15.25 kilometers, we first convert this to meters by multiplying by 1000 (since there are 1000 meters in a kilometer):
15.25 km × 1000 = 15250 meters.
Now that we know Cam rode 15250 meters and that this is 5 times the distance Dante rode, we can find Dante’s distance:
15250 meters / 5 = 3050 meters.
Dante rode 3050 meters, which is 187 meters farther than Michael did. To find the distance Michael rode, we subtract 187 meters from Dante’s distance:
3050 meters - 187 meters = 2863 meters.
Therefore, Michael rode 2863 meters.
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What is the average rate of change of the function over the interval x = 0 to x = 6? f(x)=2x−1
____
3x+5
Using the discriminant, how many real number solutions does this equation have? 3x^2 – 2 = 5x
Stephanie rdes her bike 2 miles to the post office, and then another 50 yards to the train station. How many yards long was her trip?
A bus covered the 400-km distance between points a and b at a certain speed. on the way back the bus traveled at the same speed for 2 hours and then increased the speed by 10 km/hour until it reached a, thus spending 20 fewer minutes on the return trip. how long did the return trip take?
The speed going to B is x, the time t and the distance, 400 km.
xt=400, x kph and t in hours
on the way back, the 2x+(x+10)(t-7/3), where it was total time minus 2 hours already driven minus 1/3 of an hour, the 20 minutes.
2x+xt-(7/3)x+10t-70/3=400
2x+400-(7/3)x+10t-70/3=400, 400 cancels, t=400/x
-(1/3)x+4000/x=70/3
multiply by x
-(1/3)x^2-(70/3)x+4000=0
x^2+70x-12000=0
x=(1/2)(-70+/- sqrt 52900); sqrt term=230
x=80 kph
took 5 hours to go there
on way back 2 hours at 80 kph=160 km
2h40m at 90kph=90*8/3=240 km
They add to 400 km.
The return trip took 4h40m