Step-by-step explanation:
To solve this question I would use the sin rule.
The sin rule states that
[tex] \frac{a}{ \sin(a) } = \frac{b}{ \sin(b) } [/tex]
Therefore if you substitute in your numbers you get:
[tex] \frac{a}{ \sin(90) } = \frac{15}{ \sin(60) } [/tex]
If you rearrange that you get:
[tex]a = \frac{15}{ \sin(60) } \times \sin(90) [/tex]
Therefore a = 17.3 Inches (to 3 sf)
This can also be done with basic trigonometry where you would get
[tex] \sin(60) = \frac{15}{h} [/tex]
Rearranging to
[tex]h = \frac{15}{ \sin(60) } [/tex]
meaning the answer is 13.7 inches
================================================
Work Shown:
h = unknown hypotenuse
sin(angle) = opposite/hypotenuse
sin(60) = 15/h
h*sin(60) = 15
h*sqrt(3)/2 = 15
h*sqrt(3) = 2*15
h*sqrt(3) = 30
h = 30/sqrt(3)
h = (30*sqrt(3))/(sqrt(3)*sqrt(3)
h = 30*sqrt(3)/3
h = (30/3)*sqrt(3)
h = 10*sqrt(3)
Can someone please help me with my homework page 17, 19, and 20. Thank you!
Answer:
17.B
19. C
20.C
Step-by-step explanation:
17.
[tex]\frac{m^2n^3}{p^3} \div \frac{mp}{n^2}\\=\frac{m^2n^3}{p^3} \times \frac{n^2}{mp}\\=\frac{mn^5}{p^4}[/tex]
19.
[tex]w^{2} +w-20=w^2+5w-4w-20=w(w+5)-4(w+5)=(w+5)(w-4)\\(w-3)(w-4)=w(w-4)-3(w-4)=w^2-4w-3w+12=w^2-7w+12\\reqd.~fraction ~is~ \frac{w^2-7w+12}{w^2+w-20}[/tex]
20.
[tex]\frac{w^2+5w+6}{w^2-w-12} =\frac{w^2+3w+2w+6}{w^2-4w+3w-12} =\frac{w(w+3)+2(w+3)}{w(w-4)+3(w-4)} =\frac{(w+3)(w+2)}{(w-4)(w+3)} =\frac{w+2}{w-4}[/tex]
Determine algebraically whether the function is even, odd, or neither even nor odd.
f(x) = -9 x^3 + 8x
Odd
Even
Neither
Answer: The function will be ODD.
Hope this helps! :)
QuezoMartiinez
The function f(x) = -9x³ + 8x is odd.
Option A is the correct answer.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
A function f(x) is even if f(-x) = f(x).
A function is odd if f(-x) = -f(x).
Using the above definition.
f(x) = -9x³ + 8x
Put x = -x
f(-x) = -9 x (-x³) + 8 x (-x)
f(-x) = 9x³ - 8x
f(-x) = - ( -9x³ + 8x)
f(-x) = -f(x)
This means,
The function is odd.
Thus,
f(x) is odd.
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How many different ways are there to select 24 donuts if there are 7 types of donuts available (and donuts are only distinguished by their type).?
Answer:
593,775 ways
Step-by-step explanation:
24 donuts have to be selected from 7 different varieties of donuts
n = 7
r= 24
Repetition is allowed
C(n+r-1, r) = C(7 + 24 - 1 , 24)
= C(30,24)
Recall that C(n,r) = n! /(n-r)! r!
C(30,24) = 30!/(30 - 24)! 24!
= 30!/(6!24!)
= 593,775 ways
Answer: 346,789 ways
Step-by-step explanation:
Using the formular for permutation to calculate :
P= n!/r!(n-r)!
We need to select 24 apples from 7 types of apples
n=24 ,r=7
Permutation =24!/7!(24-7)!
Permutation =6.2×10^23/1.793×10^18
P= 346 ,789 ways
Robert is Running in a race that is 3 miles long. A Distance marker is placed evenly every 1/2 mile from the beginning of the race until the end. What is the total number of distance markers on the trail?
Answer: the total number of distance markers on the trail is 6
Step-by-step explanation:
The total distance that Robert covers in the race is 3 miles.
If a distance marker is placed evenly every 1/2 mile from the beginning of the race until the end, then the total number of distance markers on the trail would be
3/0.5 = 6
The sum of two numbers is thirty seven. Using x to represent the larger of two numbers, translate "the difference between nine more than twice the larger number and the sum of the smaller number and three" into a variable expression
Answer:
Step-by-step explanation:
Let x represent the larger number.
Let y represent the smaller number.
The sum of two numbers is thirty seven. This means that the smaller number would be 37 - x
The translation into a variable expression for the the difference between nine more than twice the larger number and the sum of the smaller number and three would be
2x + 9 - (37 - x + 3)
Opening the parenthesis and putting the constants together, it becomes
2x + 9 - 37 + x - 3
2x + x + 9 - 37 - 3
3x - 31
I need help with 2b.
The topic is rational equations and restrictions are 2 and 0.
As my answer I got 0 = 8 but I’m unsure if that’s correct...
Answer:
The answer to your question
a) Values excluded = -2, 0
b) solutions = -2/3, 2
Step-by-step explanation:
Equation
[tex]\frac{4x}{2x + 4} = \frac{x+ 2}{2x}[/tex]
a) x is restricted when the denominators equal zero, so let's get them.
2x + 4 = 0 2x = 0
2x = - 4 x = 0 / 2
x = -4 / 2 x = 0
x = -2
x must be different to -2, 0
b) [tex]\frac{4x}{2x + 4} = \frac{x+ 2}{2x}[/tex]
[tex]\frac{2x}{x + 2} = \frac{x + 2}{2x}[/tex]
4x² = (x + 2)²
4x² = x² + 4x + 4
4x² - x² - 4x - 4 = 0
3x² - 4x - 4 = 0
3x² - 6x + 2x - 4 = 0
3x(x - 2) + 2(x - 2) = 0
(x - 2)(3x + 2) = 0
x₁ - 2 = 0 3x₂ + 2 = 0
x₁ = 2 x₂ = -2/3
The Rivera family went out to eat at a restaurant to celebrate a birthday. The bill for the meal, including tax, came to $126.50. They left a tip in the amount of 18% of the bill. How much did the Rivera family pay for the meal, including tip?
Answer:
$149.27
Step-by-step explanation:
18% of $126.50 is $22.77 which then added to 126.50=$149.27
A gardener is mowing a 20 by 40 yard rectangular pasture using a diagonal pattern. He mows from one corner of the pasture to the corner diagonally opposite. What is the length of this pass with the mower? Give your answer in simplified form. Recall from the Pythagorean Theorem that, for a right triangle, the square of the length of the diagonal is equal to the sum of the squares of the lengths of the sides.
A. 10 times square root 20
B. 20 times square root 2
C. 400 times square root 5
D. 20 times square root 5
20 * (sqrt) 5 is the correct answer or “D”
recently took this trust me 100% i’m just here to help anybody :)
Answer:
d
Step-by-step explanation:
using Pythagorean theorem you would do a^2 + b^2 = c^2 so 20^2 + 40^2 = 2000 do the square root of 2000 and you get 20* square-root of 5
identify the image of triangle XYZ for a composition of a 30° rotation and a 330° rotation both about point z
Answer:
As
30° + 330° = 360°
So, the figure has rotated one complete revolution as
Step-by-step explanation:
Considering the image of triangle XYZ for a composition of a
30° rotationand a
330° rotation both about point zAs
30° + 330° = 360°
It means, the figure has rotated one complete revolution. And now the figure is very much back to starting position.
Keywords: image, triangle, rotation, revolution
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Answer: The first image is the triangle XYZ and the second is the composition of 30° and 330° (360°) as a rotation about point z
does anyone know the answer??
You have to find x.
Answer:x = 9.01
Step-by-step explanation:
The given triangle is a right angle triangle.
From the given right angle triangle the hypotenuse of the right angle triangle is 17
With 35 degrees as the reference angle,
x represents the adjacent side of the right angle triangle.
The unlabelled side represents the opposite side of the right angle triangle.
θ = 35 degrees
To determine x, we would apply trigonometric ratio
Cos θ = adjacent side/hypotenuse side. Therefore,
Cos 35 = x/11
x = 11 Cos 35 = 11 × 0.8192
x = 9.01
Answer:
9.01
Step-by-step explanation:
The right angle triangle with a focused angle of degree is easily solved using Trig Ratio that can easily be recalled with this word :
SohCahToa.
What it means is:
If the Opposite side to the focused angle is given and also the hypotenuse is given, we use sine
Cosine if Adjacent and Hypotenuse is given.
Tangent if the opposite and Adjacent is given.
Now, to the question:
The Adjacent and the Hypotenuse is given. Therefore we'll use Cosine.
Therefore:
Cos 35 = x / 11
Cross multiplying:
11 cos 35 = x
11 * 0.8192 = x
Theresfore,
x = 9.01
Hope it helped?
A _______ variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure. A ▼ relative key probability random variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.
Answer:
random variable
Step-by-step explanation:
Random variable also known as stochastic variables are variables that are used in statistics and probability, The value of a random variable must be determined by chance which remain uncertain and depends on the outcome of a random process or experiment.
Unlike algebra, when random variable is introduced into a mathematical model, it is represented by capital letter and it has a single numerical value.
Scott and Tom rent a boat at Stow Lake. They start at 10:15 and end at 11:45. The boat rental costs $1.50 for every 15 minutes. How much will they pay?
Answer:they would pay a total amount of $9
Step-by-step explanation:
Scott and Tom rent a boat at Stow Lake. They start at 10:15 and end at 11:45. The number of hours for which they rented the boat would be
11:45 - 10:15 = 1 hour 30 minutes = 1.5 hours.
Converting to minutes, It becomes
60 + 30 = 90 minutes.
The boat rental costs $1.50 for every 15 minutes. Therefore, the total amount of money that they would pay is
90/15 × 1.5 = $9
Sam bought 2 granola bars and Haley Bought 5 granola bars each granola bar is the same price right in expression for the total price of the granola bars then simplify the expression to create an equivalent expression
Answer:
Step-by-step explanation:
figure out the answer by figuring out how much each bar is
Answer:
2b + 5b = b(2+5) = 7b
Step-by-step explanation:
Have a good day :)
Sam 2 granola bars
ADD
Haley 5 granola bars
Equal 7 granola bars
Which angles are coterminal with an angle drawn in standard position measuring 282°?
Select all correct angle measures.
−438∘
−78∘
78°
572°
642°
Answer:
−438°, -78°, 642°
Step-by-step explanation:
Given angle:
282°
To find the co-terminal angles of the given angle.
Solution:
Co-terminal angles are all those angles having same initial sides as well as terminal sides.
To find the positive co-terminal of an angle between 360°-720° we will add the angle to 360°
So, we have: [tex]282\°+360\°=642\°[/tex]
To find the negative co-terminal of an angle between 0° to -360° we add it to -360°
So, we have: [tex]282\°-360\°=-78\°[/tex]
To find the negative co-terminal of an angle between -360° to -720° we add it to -720°
So, we have: [tex]282\°-720\°=-438\°[/tex]
Thus, the co-terminal angles for 282° are:
−438°, -78°, 642°
The correct coterminal angles with 282° are -78° option(2) and 642° option(5). The angles -438°, 78°, and 572° are not coterminal with 282°.
In mathematics, coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles for a given angle, we add or subtract multiples of 360° (a full rotation). For the angle 282°:
1. Subtracting 360°:
⇒ 282° - 360° = -78°
2. Adding 360°:
⇒ 282° + 360° = 642°
Therefore, the angle measures -78° and 642° are coterminal with 282°. The other values, -438°, 78°, and 572°, are not coterminal with 282° as they do not share the same position after full rotations.
In summary, the correct coterminal angles with 282° are -78° and 642°.
Complete question:
Which angles are coterminal with an angle drawn in standard position measuring 282°?
Select all correct angle measures.
−438°−78°78°572°642°Solve for x.
x−11−−−−−√=5
A 14
B 16
C 25
D 36
The value of x in the expression will be 36.
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.
Given that;
The expression is;
⇒ √x - 11 = 5
Now,
Solve the expression for x as;
⇒ √x - 11 = 5
Squaring on both side;
⇒ (√x - 11)² = 5²
⇒ x - 11 = 25
Add 11 both side;
⇒ x - 11 + 11 = 25 + 11
⇒ x = 36
Thus, The value of x in the expression will be 36.
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The correct answer is option D. 36.
We need to solve the equation:
√(x - 11) = 5
To solve this, follow these steps:
Square both sides to eliminate the square root.
√(x - 11)2 = 52
x - 11 = 25
Isolate x by adding 11 to both sides:
x - 11 + 11 = 25 + 11
x = 36
So, the correct answer is 36 (option D).
Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth. x2 – 6 = x
A. x = 2, 3
B. x = –2, 3
C. x = 2, –3
D. x = –2, –3.
Answer:
3, -2
Step-by-step explanation:
If x2 + y2 = 25, what is the value of
at the point (4,3)?
O A. -25/27
OB.-7/27
OC. 7/27
D. 3/4
OE. 25/27
Answer:
A. -25/27
Step-by-step explanation:
Given:
The equation is given as:
[tex]x^2+y^2=25[/tex]
To find: [tex]\frac{d^2 y}{dx^2}[/tex] at (4, 3)
Differentiating the above equation with respect to 'x', we get:
[tex]\frac{d}{dx}(x^2+y^2)=\frac{d}{dx}(25)\\\\2x+2yy'=0\\\\x+yy'=0\\\\yy'=-x\\\\y'=\frac{-x}{y}------- (1)[/tex]
Value of [tex]y'[/tex] at (4,3) is given as:
[tex]y'_{(4,3)}=-\dfrac{4}{3}-------- (2)[/tex]
Now, differentiating equation (1) with respect to 'x' again, we get:
[tex]y''=\frac{d}{dx}(\frac{-x}{y})\\\\y''=\frac{y(-1)-(-x)y'}{y^2}\\\\y''=\frac{-y+xy'}{y^2}[/tex]
Now, value of [tex]y''[/tex] at (4,3) is given as by plugging 4 for 'x', 3 for 'y' and [tex]\frac{-4}{3}[/tex] for [tex]y'[/tex]
[tex]y''_{(4,3)}=\frac{-3+(4)(-\frac{4}{3})}{3^2}\\\\y''_{(4,3)}=\frac{-3-\frac{16}{3}}{9}\\\\y''_{(4,3)}=\frac{-9-16}{3}\div 9\\\\y''_{(4,3)}=\frac{-25}{3}\div 9\\\\y''_{(4,3)}=\frac{-25}{3}\times \frac{1}{9}\\\\y''_{(4,3)}=-\frac{25}{27}[/tex]
Therefore, the value of the second derivative at (4, 3) is option (A) which is equal to -25/27.
Need a little help with one please
Answer:
That would be the side-side-side (SSS) postulate, which states that if all the sides of a triangle are in a fixed ratio to all the corresponding sides of another triangle, then the two triangles are said to be congruent.
Step-by-step explanation:
Looking at triangles ABC and DEF, we notice that:
[tex]\frac{AB}{DE} = \frac{AC}{DF}=\frac{BC}{EF}[/tex]
since
[tex]\frac{3}{18} = \frac{6}{36}=\frac{7}{42}=\frac{1}{6}[/tex]
Let me know if you have further questions.
A manufacturing company with 450 employees begins a new project line and must increase their number of employees by 18% how many total employees does the company have now
Answer: the company now have 531 employees.
Step-by-step explanation:
The initial number of employees that the company had was 450.
The company began a new project line and must increase their number of employees by 18% . it means that the number of new employees that the company employed would be
18/100 × 450 = 0.18 × 450 = 81
Therefore, the total number of employees that the company have now would be
450 + 81 = 531
Find the value of x.
logx 8 = 0.5
А. 4
В. 16
C.32
D. 64
Answer:
The answer to your question is letter D. x = 64
Step-by-step explanation:
Equation
log x 8 = 0.5
Express the log as a exponential equation
x[tex]x^{0.5}[/tex] = 8
Express the power as a fraction
[tex]x^{1/2} = 8[/tex]
[tex]\sqrt{x^{2}}[/tex]
Look for a number which squared root is 8
[tex]\sqrt{64} = 8[/tex]
Then
x = 64
Answer:
Please mark as brainliest :)
Step-by-step explanation:
Solve the system using elimination.
3x – 4y = 9
–3x + 2y = 9
(–27, –9)
(3, 9)
(–3, –6)
(–9, –9)
Answer:
(-9,-9)
Step-by-step explanation:
Answer:
(-9,-9)
Step-by-step explanation:
A 400-room property that assigns 16 rooms to one section housekeeper and groups the section housekeepers in teams of 5 will need the following number of teams to service the guestrooms when the property is full:
Answer:
We conclude that we need 5 teams to service the guestrooms when the property is full.
Step-by-step explanation:
When the property of 400 rooms is full, and as we know that one housekeeper gets 16 rooms. We calculate to need 400/16=25 housekeeper.
We know that one team has 5 housekeepers, and that we have 25 housekeepers. We conclude that we need 25/5=5 teams to service the guestrooms when the property is full.
Assume that the mean of a normal distribution of IQ scores is 102 and the standard deviation is 15. You have been told that your test score is 1 standard deviation above the mean. What is your IQ test score?
Answer: your IQ test score is 117
Step-by-step explanation:
Assume that the mean of a normal distribution of IQ scores is 102 and the standard deviation is 15.
If you have been told that your test score is 1 standard deviation above the mean, it means that your score would be
102 + 15 = 117
According to the empirical rule, 68% of data falls within the first standard deviation from the mean. This means that 117 falls within 68% of the scores.
Simplify the following expressions by combining like terms.
10z-11z+2-5=
Answer:
-1z-3
Step-by-step explanation:
10z-11z = -1z
+2-5= -3
= -z-3
Answer:
-z - 3
Step-by-step explanation:
10z - 11z + 2 - 5
Simplifying:
10z - 11z = -z
+2 - 5 = -3
Theresfore; 10z - 11z +2 - 5 = -z - 3
A rectangular box 4 meters long, 3 meters wide and 2 meters tall. On the box is a cat, and the box is floating in water such that half the box is under water. The density of the box is 300 kg/m³. What is the mass of the cat?
Answer:
Mass of the cat = 16800kg
Step-by-step explanation:
The volume of the box = its length * its breath * its height- 4m × 3m × 2m = 24[tex]m^{3}[/tex]
The standard density of water is 1000 Kg/[tex]m^{3}[/tex]
By Archimedes principle, the mass of a body floating body is equivalent to the mass of the volume liquid displaced
in this case we have
Mass of water displaced = Density of the water × Volume of the water displaced = 1000 Kg/[tex]m^{3}[/tex] × 24[tex]m^{3}[/tex] = 24000kg
The mass of the box = Box density × Box volume = 24[tex]m^{3}[/tex] × 300kg/[tex]m^{3}[/tex] = 7200 kg Hence the mass of water displaced = mass of the foating box + mass of th cat
24000kg = mass of cat +7200kg
mass of cat = 24000kg - 7200kg = 16800kg
The Hall family and the Nguyen family each used their sprinklers last summer. The water output rate for the Hall family's sprinkler was 35L per hour. The water output rate for the Nguyen family's sprinkler was 15L per hour. The families used their sprinklers for a combined total of 50 hours, resulting in a total water output of 1050L. How long was each sprinkler used?
Answer:
Nguyen family uses sprinkler for 35 hours while the Hall family uses sprinkler for 15 hours
Step-by-step explanation:
Let h and n be the number of hours that the Hall and Nguyen family individually use their sprinkler for, respectively. Since they use for a combined total of 50 hours, h + n = 50, or h = 50 - n
Also their total water output is 1050 L, we can have the following equation
35h + 15n = 1050
Substitute h = 50 - n and we have
35(50 - n) + 15n = 1050
1750 - 35n + 15n = 1050
35n - 15n = 1750 - 1050
20n = 700
n = 35 hours
h = 50 - n = 50 - 35 = 15 hours
need help asap
Solve the equation by completing the square
-2x^2+x=0
a. {0,-0.5}
b. {1,0}
c. {0.5,0}
d. {0}
Answer:
Option C) [tex]{\{0.5,0}\}[/tex] is correct
The solution to the given equation is [tex]{\{0.5,0}\}[/tex]
Step-by-step explanation:
Given quadratic equation is [tex]-2x^2+x=0[/tex]
To solve the equation by completing the square :
[tex]-2x^2+x=0[/tex]
[tex]-2(x^2-\frac{x}{2})=0[/tex]
[tex]x^2-\frac{x}{2}=0[/tex]
Rewritting the above equation as below :
[tex]x^2-2(x)\frac{1}{4}+\frac{1}{4^2}-\frac{1}{4^2}=0[/tex]
[tex](x-\frac{1}{4})^2-\frac{1}{4^2}=0[/tex]
[tex](x-\frac{1}{4})^2=\frac{1}{4^2}[/tex]
Taking square root on both sides we get
[tex]\sqrt{(x-\frac{1}{4})^2}=\sqrt{\frac{1}{4^2}}[/tex]
[tex](x-\frac{1}{4})=\pm\frac{1}{4}[/tex]
[tex]x=\pm\frac{1}{4}+\frac{1}{4}[/tex]
[tex]x=+\frac{1}{4}+\frac{1}{4}[/tex] and [tex]x=-\frac{1}{4}+\frac{1}{4}[/tex]
[tex]x=\frac{2}{4}[/tex] and [tex]x=0[/tex]
Therefore [tex]x=\frac{1}{2}[/tex] and x=0
[tex]x=0.5[/tex] and x=0
Therefore the solution to the given equation is [tex]{\{0.5,0}\}[/tex]
Option C) [tex]{\{0.5,0}\}[/tex] is correct
A 28ft ladder leans against a wall so that the base of the ladder is 5ft away from the base of the wall. How far up the wall does the ladder reach?
Round to the nearest tenth.
Answer:
the top of the ladder is 27.5ft above the ground.
Step-by-step explanation:
The ladder makes an angle, θ with the ground thus forming a right angle triangle with the wall of the house.
The length of the ladder represents the hypotenuse of the right angle triangle.
The ground distance from the base of the wall to the foot of the ladder represents the adjacent side of the right angle triangle.
Therefore, to determine how high up is the top of the ladder on the wall, x, we would apply Pythagoras theorem which is expressed as
Hypotenuse² = opposite side² + adjacent side²
28² = x² + 5²
784 = x² + 25
x² = 784 - 25 = 759
x = √759 = 27.5ft
Using the Pythagorean theorem, the ladder reaches approximately 27.5 feet up the wall when the base is 5 feet from the wall.
Explanation:The question involves finding the height up a wall that a 28ft ladder reaches when its base is 5ft from the wall. To solve this, we will use the Pythagorean theorem which relates the sides of a right-angled triangle. The ladder, wall, and ground form a right-angled triangle with the ladder as the hypotenuse and the wall and ground as the other two sides.
Let h be the height the ladder reaches up the wall. Using the Pythagorean theorem, we have:
28² = h² + 5²
Solving for h:
28² - 5² = h²
784 - 25 = h²
759 = h²
h ≈ √759
h ≈ 27.5 (to the nearest tenth)
Therefore, the ladder reaches approximately 27.5 feet up the wall.
If y = e^5t is a solution to the differential equation d^2 y/dt^2 - 13 dy/dt + ky = 0, find the value of the constant k and the general form y = Ae^5t + Be^at of the solution to the above equation, (i.e. find a). (Use constants A, B, etc., for any constants in your solution formula.)
Answer:
k = -12/5
A = 125/12
B = -325/12
a = 5
Step-by-step explanation:
y = e^5t
Dy/dt = 5e^5t
d2y/dt2 = 25e^5t
Inputting the values of dy/dt and d2y/dt2 into the equation above, we have:
25e^5t - 13e^5t + 5k(e^5t) = 0
12e^5t + 5k(e^5t) = 0
e^5t(12 + 5k) = 0
12 + 5k = 0
k = -12/5
The equation becomes,
d2y/dt2 - 13dy/dt -12/5y = o
So rearranging the equation,
-5/12d2y/dt2 + 65/12dy/dt + y = 0
y = 5/12(25e^5t) - 65/12(5e^5t)
y = 125/12e^5t - 325/12e^5t
Therefore,
k = -12/5
A = 125/12
B = -325/12
a = 5
The value of the constant k is 40. The general form of the solution is y = A[tex]e^{5t}[/tex] + B[tex]e^{8t}[/tex]
Let's start by recognizing that if y = [tex]e^{5t}[/tex] is a solution to the differential equation d²y/dt² - 13 dy/dt + ky = 0, we need to find the value of the constant k and the general form y = A[tex]e^{5t}[/tex] + B[tex]e^{at}[/tex] of the solution. To do this, we need to determine k and a.
1. First, calculate the first and second derivatives of y = [tex]e^{5t}[/tex]
First derivative: dy/dt = 5[tex]e^{5t}[/tex]Second derivative: d²y/dt² = 25[tex]e^{5t}[/tex]2. Substitute these into the differential equation:
d²y/dt² - 13 dy/dt + ky = 0
3. Substituting, we get:
25[tex]e^{5t}[/tex] - 13(5[tex]e^{5t}[/tex]) + k[tex]e^{5t}[/tex] = 0
25[tex]e^{5t}[/tex] - 65[tex]e^{5t}[/tex] + k[tex]e^{5t}[/tex]= 0
(25 - 65 + k)[tex]e^{5t}[/tex] = 0
(-40 + k)[tex]e^{5t}[/tex] = 0
4. For this to hold true, the following must be true:
k - 40 = 0
Thus:
k = 40
The general solution to the differential equation can be expressed as:y = A[tex]e^{5t}[/tex] + B[tex]e^{at}[/tex]
1. To find a, substitute y = [tex]e^{at}[/tex] into the equation:
d²([tex]e^{at}[/tex])/dt² - 13 d([tex]e^{at}[/tex])/dt + 40[tex]e^{at}[/tex] = 0
We get:
a²[tex]e^{at}[/tex]- 13a[tex]e^{at}[/tex] + 40[tex]e^{at}[/tex]= 0
(a² - 13a + 40)[tex]e^{at}[/tex] = 0
2. The characteristic equation is:
a² - 13a + 40 = 0
3. Solve for a using the quadratic formula:
a = [13 ± √(13² - 4⋅40)] / 2
a = [13 ± √(169 - 160)] / 2
a = [13 ± √9] / 2
a = [13 ± 3] / 2
4. The roots are:
a = 8a = 5Since y = [tex]e^{5t}[/tex] is a solution already, the other root a = 8 is the additional solution. Thus, the general solution to the differential equation is:
y = A[tex]e^{5t}[/tex] + B[tex]e^{8t}[/tex].
Jamal's Seed Emporium claims that 75% of its lily seeds will germinate. Suppose the company's claim is true. Sierra buys a packet with 25 lily seeds from Jamal's Seed Emporium and plants them in her garden. What is the probability that exactly 18 seeds will germinate?
Answer:
P = 0.1654
Step-by-step explanation:
Binomial probability:
P = nCr pʳ qⁿ⁻ʳ
where n is the number of trials,
r is the number of successes,
p is the probability of success,
and q is the probability of failure (1−p).
P = ₂₅C₁₈ (0.75)¹⁸ (0.25)²⁵⁻¹⁸
P = 480,700 (0.75)¹⁸ (0.25)⁷
P = 0.1654