The correct answers are x³ + 19 x² + 103 x - 135 = 0 and x³ + 19 x² + 103 x - 735 = 0.
Given:
A box with dimensions 3 cm × 5 cm × 11 cm
Current volume of given box = 3×5×11 = 165 cm^3
New volume of box = (3+x) × (5+x) × (11+x)
= (15 + 8 x + x²) × (11+x)
= 165 + 103 x + 19 x^2 + x³
For volume to be 300 cm³
=> x³+ 19 x² + 103 x + 165 = 300
=> x³ + 19 x² + 103 x - 135 = 0
For volume to be 900 cm³ -
=> x³ + 19 x² + 103 x + 165 = 900
=> x³ + 19 x²+ 103 x - 735 = 0
Therefore, x^3 + 19 x^2 + 103 x - 135 = 0 and x³ + 19 x² + 103 x - 735 = 0 are the solution.
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Three points A, B and C lie on a level plane. B is 7 km from A on a bearing of 030°. C is 5 km from B on a bearing of 280°. The distance AC is Answer
km. The bearing of A from C is Answer
°. The area of the triangle ABC is Answer
km2
The distance AC is approximately 6.47 km, the bearing of A from C is approximately 313.8°, and the area of triangle ABC is approximately 14.8 km².
To solve this problem, we can use the law of cosines to find the length of AC and the law of sines to find the angle at A.
First, we can use the given information to draw a diagram and label the angles and sides:
C
/ \
/ \
5 / \ 7
/ \
/ \
A /___θ_____\ B
From the given information, we know that:
AB = 7 km
BC = 5 km
∠ABC = 100° (since ∠ABD = 180° - 30° - 70° = 80° and ∠DBC = 180° - 280° = 100°)
Using the law of cosines, we can find the length of AC:
AC^2 = AB^2 + BC^2 - 2ABBCcos(∠ABC)
AC^2 = 7^2 + 5^2 - 2(7)(5)cos(100°)
AC^2 ≈ 41.84
AC ≈ 6.47 km
Next, we can use the law of sines to find the angle at A:
sin(∠CAB)/AC = sin(∠ABC)/AB
sin(∠CAB)/6.47 = sin(100°)/7
sin(∠CAB) ≈ 0.276
∠CAB ≈ 16.2°
To find the bearing of A from C, we can use the fact that the bearing is the angle measured clockwise from north. Since ∠CAB is acute and the bearing of B from A is 30° east of north, the bearing of A from C is:
360° - (30° + ∠CAB) ≈ 313.8°
Finally, to find the area of triangle ABC, we can use the formula:
Area = (1/2)ABBCsin(∠ABC)
Substituting the given values, we get:
Area = (1/2)(7)(5)sin(100°)
Area ≈ 14.8 km²
Therefore, the distance AC is approximately 6.47 km, the bearing of A from C is approximately 313.8°, and the area of triangle ABC is approximately 14.8 km².
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Find an integer C that will make the polynomial factorable 32 − 8 + C
HELPPPP I NEED THIS ASAP
We can factor the polynomial as:-32a^2 - 82a + 31 = 32(a - \frac{1}{2})(a - \frac{31}{32})
.To make the polynomial 32a^2 - 82a + C factorable, we need to find an integer value for C such that the quadratic expression can be factored into two binomials.
To do this, we can use the formula for the discriminant: b^2 - 4ac. Since we want the discriminant to be a perfect square, we can set it equal to some integer k^2 and solve for C:
82^2 - 4(32)(C) = k^2
6724 - 128C = k^2
We can try different values of k and solve for C until we find an integer solution. For example, if we let k = 10, we get:
10^2 = 6724 - 128C
C = 31
To verify that this value of C works, we can factor the trinomial using the X formula:
a = 32, b = -82, c = 31
X = (-b ± sqrt(b^2 - 4ac)) / 2a
X = (82 ± sqrt(82^2 - 4(32)(31))) / 2(32)
X = (82 ± 8) / 64
So, the roots of the quadratic are:
a = 32, X1 = 1/2, X2 = 31/32
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1. Before you took this course, you probably heard many stories about Statistics courses. Oftentimes
parents of students have had bad experiences with Statistics courses and pass on their anxieties to their
children. To test whether actually taking AP Statistics decreases students' anxieties about Statistics, an
AP Statistics instructor gave a test to rate student anxiety at the beginning and end of his course.
Anxiety levels were measured on a scale of 0-10. Here are the data for 16 randomly chosen students
from a class of 180 students:
paired
t
SU
Pre-course anxiety level
Post-course anxiety level
Difference (Post - Pre)
7 6 9 5
4 3 7 3
-3 -3 -2 -2
6 7
4 5
-2 -2
5
4
-1
7 6 4
6 5 3
-1 -1 -1
3
2
-1
2 1
2 1
0 0
3
3
0
4
4
0
2
3
1
The assumptions include:
1. The observations are normally distributed.
2 The observertions are independent.
What are the test hypothesis?Test hypotheses include the null hypothesis that the anxiety level is same before after the course and the alternative hypothesis that the anxiety level derveases after taking the course.
The standard deviation is 1.1475. The value of to will be:
= -1.125 / (1.1475 / √16)
= 3.9212
Based on the p value, it should be noted that the null hypothesis is rejected.
In conclusion, the anxiety level decrease after the students take the course.
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Before you took this course, you probably heard many stories about Statistics courses. Oftentimes parents of students have had bad experiences with Statistics courses and pass on their anxieties to their children. To test whether actually taking AP Statistics decreases students' anxieties about Statistics, an AP Statistics instructor gave a test to rate student anxiety at the beginning and end of his course.
Anxiety levels were measured on a scale of 0-10. Here are the data for 16 randomly chosen students from a class of 180 students:
Do the data indicate that anxiety levels about Statistics decreases after students take AP Statistics? Test an appropriate hypothesis and state your conclusion.
A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model:
E(y) = ?0 + ?1x,
where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were obtained:
? = 74.80 + 19.72x
What are the properties of the least squares line, ? = 74.80 + 19.72x?
For each additional room in the house, we estimate the appraised value to increase $74,800.
We estimate the base appraised value for any house to be $74,800.
For each additional room in the house, we estimate the appraised value to increase $19,720.
There is no practical interpretation, since a house with 0 rooms is nonsensical.
The properties of the least squares line are that the intercept represents the estimated base appraised value for any house in East Meadow, and the slope represents the estimated increase in appraised value for each additional room in the house.
The properties of the least squares line, ? = 74.80 + 19.72x, are as follows:
1. The intercept, ?0, is 74.80. This represents the estimated base appraised value for any house in East Meadow, regardless of the number of rooms. However, there is no practical interpretation for this value, since a house with 0 rooms is nonsensical.
2. The slope, ?1, is 19.72. This represents the estimated increase in appraised value for each additional room in the house. For each additional room in the house, we estimate the appraised value to increase $19,720.
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Find the equation of the line shown.
y
10
655521 LEO
O
1 2 3 4 5 6 7 8 9 10
X
The equation of the line shown is y = 1x + 9.
What is the equation?An equation is a statement of equality between two expressions. It is typically expressed in mathematical notation, with the left side of the equation equal to the right side. Equations are used to describe how two values are related, and can be used to solve problems or predict outcomes. Equations can be used in any field of study, from physics to economics.
The equation of the line shown can be found by using the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is m = (10 - 9)/(10 - 0) = 1.
The y-intercept is b = 9, as this is the point at which the line crosses the y-axis. Therefore, the equation of the line shown is y = 1x + 9.
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Complete questions as follows-
this is happening do proper format and calculation
Find the equation of the line shown.
y
10
S87654321
9
0 1 2 3 4 5 6 7 8 9 10
X
The border of a susan b. Anthony dollar is in the shape of a regular polygon. How many sides does the polygon have what is the measure of each angle of the border round your answer to the nearest degree
The polygon has 12 sides and each angle has a measure of 30°.
The border of a Susan B. Anthony dollar is in the shape of a regular polygon. A regular polygon is a polygon with equal sides and equal angles. To calculate the number of sides and measure of the angles, the formula for finding the measure of the interior angles of a regular polygon is used. The formula is (n-2)180/n where n is the number of sides. For the Susan B. Anthony dollar, the number of sides is 12, so the measure of the interior angles is (12-2)180/12 = 30°. Therefore, the polygon has 12 sides and each angle has a measure of 30°, rounded to the nearest degree.
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Wildgrove is 7 miles due north of the airport, and Yardley is due east of the airport. If the distance between Wildgrove and Yardley is 9 miles, how far is Yardley from the airport? If necessary, round to the nearest tenth.
Please respond soon.
Use the Pythagorean theorem.
Thank you!
Answer:
Yardley is approximately 5.7 miles away from the airport.
Step-by-step explanation:
Let's use the following variables:
x = distance between Yardley and the airport
Using the Pythagorean theorem:
x^2 + 7^2 = 9^2
x^2 + 49 = 81
x^2 = 32
x = sqrt(32)
x ≈ 5.7
Therefore, Yardley is approximately 5.7 miles away from the airport.
Answer:
5.7 miles
Step-by-step explanation:
To find:-
The distance between airport and Yardley.Answer:-
We are given that airport and Wildgrove are 7miles apart and the distance between Yardley and Wildgrove is 9miles . We are interested in finding out tge distance between Yardley and airport.
As you can see from the figure attached, the distance can be calculated using Pythagoras theorem ,
[tex]\rm\implies a^2+b^2 = h^2 \\[/tex]
where ,
h is the longest side of the triangle (hypotenuse).a and b are two other sides .Here the longest side is 9 miles and one other side is 7 miles . On substituting the respective values, we have;
[tex]\rm\implies a^2 + (7)^2 = 9^2 \\[/tex]
[tex]\rm\implies a^2 + 49 = 81 \\[/tex]
[tex]\rm\implies a^2 = 81 - 49 \\[/tex]
[tex]\rm\implies a^2 = 32 \\[/tex]
[tex]\rm\implies a =\sqrt{32} \\[/tex]
[tex]\rm\implies a = 5.65 \\[/tex]
[tex]\rm\implies \red{ a = 5.7 } \\[/tex]
Hence the distance between Yardley and airport is 5.7 miles .
Kohli scored 10 runs less than Rohit in innings. Rahul scored 5 runs more than Rohit. In total they scored 442 runs. What was the score of kohli
Throughout his innings, Kohli scored 432 runs in total.. Rohit scored 422 runs and Rahul scored 427 runs. In total, they scored 442 runs.
Kohli, Rohit and Rahul are three of the greatest batsmen in Indian cricket. In a match, they scored 442 runs in total.Rohit scored the most runs (422 total),. Kohli scored 10 runs less than Rohit, making his total score 432 runs. Rahul was the third highest scorer, with 5 runs more than Rohit, making his total score 427 runs. This shows the great batting prowess of the three batsmen and how they were able to score a combined 442 runs between them. This goes to show the great talent and skill that the three players possess and how they are able to make such a great contribution to their team's total score.
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If point X (-4,5) was rotated 180 Degrees counterclockwise around the origin, what would the coordinates of X' be?
Answer:
(4,-5)
Step-by-step explanation:
(x,y) → (-x, -y)
(-4,5) → (4,-5)
Helping in the name of Jesus.
I really need help with this math problem
Step-by-step explanation:
The area with the highest inches receives most rainfall 1¼
The area with the lowest inches received lowest rainfall 1/8
1¼ location will have highest frequency using the ⅛line plot. There are 10 of ⅛ in 1¼
Add all the inches
1/8 + 3/8 +1/8 + 1/8 + 3/4+ 3/4 + 1/4 + 1¼ + 1/4 +1
6/8 + 2¼ + 1
6/8 + 3¼
6/8 + 13/4 = 32/8
Total rain = 4inches
Find the maximum value of the objective function and the values of x and y for which it occurs.
F=5x+2y
x+2y<6. x>0 and y>0
2x+y<6
Answer: o find the maximum value of the objective function F=5x+2y and the values of x and y for which it occurs, we need to graph the constraints and determine the feasible region. Then, we can evaluate the objective function at the vertices of the feasible region to find the maximum value.
First, we will graph the constraints x+2y<6 and 2x+y<6. To do this, we can rewrite the inequalities as equations and graph the corresponding lines:
x+2y=6 (solid line)
2x+y=6 (dashed line)
Next, we need to determine which side of each line satisfies the inequality. To do this, we can choose a point on one side of each line and substitute its coordinates into the inequality. If the inequality is true, then that side of the line satisfies the inequality. If not, then the other side satisfies the inequality.
For example, let's choose the point (0,0) as a test point for the inequality x+2y<6:
0+2(0)<6
This is true, so the side of the line on which (0,0) lies satisfies the inequality. Similarly, we can choose the point (0,0) as a test point for the inequality 2x+y<6:
2(0)+0<6
This is also true, so the side of the line on which (0,0) lies satisfies the inequality.
Now, we can shade in the feasible region, which is the region of the graph that satisfies all of the constraints. It is the region bounded by the two lines and the axes, and it is shown in the figure below:
lua
| .
| .
| .
3 | .
| .
| .
2 | .
|.
----------------
0 1 2 3 4 5 6
The vertices of the feasible region are (0,3), (2,2), and (3,0). To find the maximum value of the objective function, we evaluate it at each vertex:
F(0,3) = 5(0) + 2(3) = 6
F(2,2) = 5(2) + 2(2) = 14
F(3,0) = 5(3) + 2(0) = 15
Therefore, the maximum value of the objective function is 15, and it occurs when x=3 and y=0.
enjoy (:
The trinomial 2x2 + 13x + 6 has a linear factor of x + 6.
2x2 + 13x + 6 = (x + 6)(?)
What is the other linear factor?
pls tell me how to do the table thing
The trinomial( consists three different terms), 2x² + 13x + 6, has a linear factor of (x + 6) and the linear factor of (x+6) is equals to a (2x+1).
We have, a trinomial 2x² + 13x + 6 has a linear factor of x + 6 and we have to determine that linear factor of (x + 6). To determine the linear factor of (x + 6) either dividing the trinomial by (x+6) or factorization of this trinomial. Factoring a trinomial means expanding an equation into the product of two or more binomials. Here, middle term of trinomial, b = 13
last term, c = 6
Using spliting middle term method for factorization, 2x²+ 13x + 6 = 2x² + 12x + x + 6
=> 2x² + 13x + 6 = 2x( x + 6) + 1(x + 6 )
Common factor from binomials,
=> 2x² + 13x + 6 = (x + 6)(2x + 1)
Thus, the linear factor of (x+6) is (2x +1).
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What is the length of the hypotenuse?
Answer:
41
Step-by-step explanation:
a^2+b^2=c^2
So...
40^2+9^2=1681
√1681=41
The graph is a translation of one of the basic functions , , , . Find the equation that defines the function.
help plsss its due today
The equation of line is y = -3x + 2
Define equation of variable?An equation for two variables is a mathematical statement that relates two variables, typically represented by x and y, using mathematical symbols and operations such as addition, subtraction, multiplication, and division.
To find the equation of the line that passes through the given points (2, -4), (3, -7), (4, -10), and (5, -13), we can use the slope-intercept form of the equation of a line:
y = mx + b, where slope of the line is m and y-intercept is b.
First, let's find the slope of the line using two of the points, say (2, -4) and (3, -7). The slope, m, is given by:
m = [tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
= (-7 - (-4)) / (3 - 2)
= -3
we know the slope, we can use one of the points, say (2, -4), and the slope to find the y-intercept, b. Use the slope intercept to form the equation of a line:
y = mx + b, and substitute in the slope and coordinates of one of the points:
-4 = (-3)(2) + b
Simplifying this equation, we get:
b = -4 + 6
= 2
Therefore, the equation of the line that passes through the given points (2, -4), (3, -7), (4, -10), and (5, -13) is: y = -3x + 2
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1. Use properties of logarithms with the given approximations to evaluate the expression. Use logb2=0.693 and/or logb6=1.792 to find logb12.2. Complete parts (a) and (b). a. Write the inverse of y=8x in logarithmic form. b. Graph y=8x and its inverse and discuss the symmetry of their graphs.3. 2log2^15 EVALUATEUse properties of logarithms with the given approximations to evaluate the expression.loga3≈0.477andloga5≈0.699.Use one or both of these values to evaluate loga27.Write as a single logarithm. Assume that variables represent positive numbers.5log2x+2log2zOn the basis of data for the years 1918 through1997, the expected life span of people in a country can be described by the functionf(x)=12.576ln x+17.088 years, where x is the number of years from1905 to the person's birth year. What does this model estimate the life span to be for people born in 1941?This model estimates that the life span for people born in1941 is
1. Use properties of logarithms with the given approximations to evaluate the expression. Use log2=0.693 and/or log6=1.792 to find log12.
Answer: log12 = log6 + log2 = 1.792 + 0.693 = 2.485.
2. Complete parts (a) and (b).
a. Write the inverse of y=8x in logarithmic form.
Answer: log8(y) = x.
b. Graph y=8x and its inverse and discuss the symmetry of their graphs.
Answer: The graph of y=8x and its inverse will have the same shape, but it will be flipped across the line y = x. This is known as the symmetry of their graphs.
3. 2log215 EVALUATE Use properties of logarithms with the given approximations to evaluate the expression. log3≈0.477 and log5≈0.699. Use one or both of these values to evaluate log27. Write as a single logarithm. Assume that variables represent positive numbers. 5log2x+2log2z
Answer: log27 = log5 + log3 = 0.477 + 0.699 = 1.176.
4. On the basis of data for the years 1918 through 1997, the expected life span of people in a country can be described by the function f(x) = 12.576ln x + 17.088 years, where x is the number of years from 1905 to the person's birth year. What does this model estimate the life span to be for people born in 1941?
Answer: This model estimates that the life span for people born in 1941 is 70.698 years.
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Watch help video Given f(x)=x^(3)+kx+9, and the remainder when f(x) is divided by x-3 is 27 , then what is the value of k ?
When f(x) is divided by x-3 is 27 , then what is the value of k is -3
What is remainder theοrem?Remainder Theοrem is an apprοach οf Euclidean divisiοn οf pοlynοmials. Accοrding tο this theοrem, if we divide a pοlynοmial P (x) by a factοr ( x – a); that isn’t essentially an element οf the pοlynοmial; yοu will find a smaller pοlynοmial alοng with a remainder.
When f(x) is divided by x - 3, the remainder is 27, which means that
f(3) = 27.
We can use this fact tο sοlve fοr k as fοllοws:
f(x) = x³ + kx + 9
f(3) = 27
Substituting x = 3 intο the expressiοn fοr f(x), we get:
f(3) = 3³ + k(3) + 9 = 27
Simplifying the left side, we get:
27 + 3k + 9 = 27
Cοmbining like terms, we get:
3k + 36 = 27
Subtracting 36 frοm bοth sides, we get:
3k = -9
Dividing bοth sides by 3, we get:
k = -3
Therefοre, the value οf k is -3.
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What’s the answer to this
Answer: 45 ml/h
Step-by-step explanation: that is s/t graph
read in any point on the part of graph distance , and divide
it with time .
95:2 hours,
PREVIOUS ANSWER: PYTHAGOREAN THEOREM 28.5 Scavenger Hunt B Find the value of x. 20 15 W 8
The value of x is 25 using Pythagoras theorem.
What are angles?A pοint where twο lines meet prοduces an angle.
The breadth οf the "οpening" between these twο rays is referred tο as a "angle". It is depicted by the figure.
Radians, a unit οf circularity οr rοtatiοn, and degrees are twο cοmmοn units used tο describe angles.
By cοnnecting twο rays at their ends, οne can make an angle in geοmetry. The sides οr limbs οf the angle are what are meant by these rays.
The limbs and the vertex are the twο main parts οf an angle.
The cοmmοn terminal οf the twο beams is the shared vertex.
Pythagοras theοrem,
perpendicular²= hypοtenuse² + base²
Find the hypotenuse of lower triangle
c² = 15² + 9²
c = [tex]\sqrt{15^2 + 8^2}[/tex]
c = 17
Now, c is the base altitude of upper triangle
So
x² = [tex]\sqrt{20^2 + 15^2}[/tex]
x = 25
Thus, the value of x is 25 using Pythagoras theorem.
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Nrite equations of the lines through the given point parallel to and perpendicular to the given li x+y=3,(-3,2)
Answer:
vfejwbjelkn
Step-by-step explanation:
Tina make 480 sandwiches she only makes tuna sandwiches beef sandwiches and ham sandwiches amd cheese sandwiches 3/8 of the sandwiches are tuna 35% of the sandwiches are beef the ratio of the number of ham sandwhiches to the number of cheese sanwiches is 5:6 work out the number of ham sandwhiches that tina makes
= 5x = 5(12) = 60 sandwiches, the number of ham sandwiches. Tina makes 60 ham sandwiches.
First, we need to find the total number of each type of sandwich that Tina makes:
3/8 of the sandwiches are tuna, so the number of tuna sandwiches
= 3/8 x 480
= 180 sandwiches
35% of the sandwiches are beef, so the number of beef sandwiches
= 35% x 480
= 0.35 x 480
= 168 sandwiches
The remaining sandwiches must be ham and cheese sandwiches. Let's call the number of ham sandwiches "h" and the number of cheese sandwiches "c". We know that h:c = 5:6, which means that h = 5x and c = 6x for some common factor x. The total number of sandwiches is 480, so:
h + c = 480 - 180 - 168
h + c = 132
Substituting h = 5x and c = 6x, we get:
5x + 6x = 132
11x = 132
x = 12
Therefore, the number of ham sandwiches = 5x = 5(12) = 60 sandwiches. Tina makes 60 ham sandwiches.
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part B observe the graph at what point do the lines appear to insersect
Answer:
Step-by-step explanation:
Solve for y. Then find the values of y that correspond
to the given values of x for the linear equation.
y + 8x = -2 forx = 0, 1, 2
Answer:
no one can solve that because how you spelled it but good luck
Answer:
y= -2 when x is 0,y= -10 where x is 1 and y= -18 where x is 2
Step-by-step explanation:
when you equate x to 0 then you substitute to the equation give which is y+8x= -2, the same thing applies to the next numbers which is 1 and 2. to show it will look like is suppose to be in the form like when x is 0 it will be y+8(0)= -2
which will solve to get y+0=-2
which is the same as y=-2
2(-3x+5)+2(x+4) when x=2
Answer:
12
Step-by-step explanation:
p(x) = 2(-3x+5)+2(x+4)
Substitution :
p(2) = 2(-3(2) +5) + 2(2+5)
= 2(-6 + 5) + 2 (7)
= 2(-1) + 2(7)
= -2 + 14
= 12
Write an exponential function in the form y=ab^xy=ab
x
that goes through points (0, 4)(0,4) and (2, 324)(2,324).
According to the question the exponential function that goes through the two points is: [tex]y = 4(9^x).[/tex]
Given,
We must first determine the values of a and b before we can build an exponential function with the form y = abx that passes through the points (0,4) and (2,324).
Using the point (0,4), we have:
4 = ab^0
4 = a(1)
a = 4
Using the point (2,324), we have:
324 = 4b^2
b^2 = 81
b = 9 (since b must be positive)
So the exponential function that goes through the two points is:
y = 4(9^x)
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Reed's number cube numbered (1, 2, 3, 4, 5, 6) is rolled and a spinner with 4 sections (A, B, C, D) is spun. P(1 and A)
The probability of rolling a 1 and spinning section A on the spinner is 1/24.
To find the probability of rolling a 1 and spinning section A on the spinner, we need to first determine the total number of possible outcomes for the experiment. The number cube has 6 possible outcomes, and the spinner has 4 possible outcomes. Therefore, the total number of possible outcomes for the experiment is 6 x 4 = 24.
Next, we need to determine the number of outcomes that meet the criteria of rolling a 1 and spinning section A. Rolling a 1 has a probability of 1/6, and spinning section A has a probability of 1/4.
The probability of both events occurring simultaneously is the product of the two probabilities, which is (1/6) x (1/4) = 1/24.
This means that out of the 24 possible outcomes, only one outcome will result in rolling a 1 and spinning section A on the spinner.
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Complete Question:
A number cube numbered (1, 2, 3, 4, 5, 6) is rolled and a spinner with 4 sections (A, B, C, D) is spun. Find the probability of P(1 and A).
I tried it two times and got it wrong, please help!
Answer:
[tex]\dfrac{dy}{dx}=-\dfrac{52x\left(3x^{37}y-1\right) }{4x^{39}+9y^8}[/tex]
Step-by-step explanation:
Note: I will provide links with descriptions/steps to each of the rules used in this differential equation at the bottom of this answer.
Given
[tex]-26x^2+4x^{39} y+y^9=-21[/tex]
Find [tex]\dfrac{dy}{dx}[/tex]
Differentiate the left side of the equation.
[tex]\dfrac{d}{dx}\left(-26x^2+4x^{39}y+y^9\right)= \dfrac{d}{dx}\left(-21\right)[/tex]
Focus the left side of the equation. Apply the Sum Rule for derivatives.
[tex]\dfrac{d}{dx} \left[-26x^2\right]+\dfrac{d}{dx} \left[4x^{39}y\right]+\dfrac{d}{dx} \left[y^9\right][/tex]
Lets evaluate each derivative.
1. Evaluate [tex]\frac{d}{dx} \left[-26x^2\right][/tex]
Differentiate using the Power Rule for derivatives.
[tex]\dfrac{d}{dx} \left[-26x^2\right]=2*26x^{2-1}[/tex]
Simplify.
[tex]2*-26x^{2-1}\\-52x^{2-1}\\-52x[/tex]
2. Evaluate [tex]\frac{d}{dx} \left[4x^{39}y \right][/tex]
Differentiate using the Product Rule for derivatives.
[tex]4\left(x^{39}\dfrac{d}{dx}\left[y\right]+y\dfrac{d}{dx}\left[x^{39}\right]\right)[/tex]
Rewrite [tex]\dfrac{d}{dx}\left[y\right][/tex] as [tex]y'[/tex]
[tex]4\left(x^{39}y'+y\dfrac{d}{dx}\left[x^{39}\right]\right)[/tex]
Differentiate using the Power Rule for derivatives.
[tex]4\left(x^{39}y'+y\left(39x^{39-1} \right)\right)[/tex]
Simplify.
[tex]4\left(x^{39}y'+39yx^{38} \right)[/tex]
3. Evaluate [tex]\frac{d}{dx} \left[y^9\right][/tex]
Differentiate using the Chain Rule for derivatives.
[tex]9y^8\dfrac{d}{dx} \left[y\right][/tex]
Rewrite [tex]\dfrac{d}{dx}\left[y\right][/tex] as [tex]y'[/tex]
[tex]9y^8y'[/tex]
Add up all of the derivatives then simplify.
[tex]-52x+4\left(x^{39}y'+39yx^{38} \right)+9y^8y'[/tex]
[tex]-52x+4\left(x^{39}y'\right)+4\left(39yx^{38} \right)+9y^8y'[/tex]
[tex]-52x+4x^{39}y'+156yx^{38}+9y^8y'[/tex]
[tex]4x^{39}y'+156x^{38}y+9y^8y'-52x[/tex]
Focus the right side of the equation.
[tex]\dfrac{d}{dx}\left(-21\right)[/tex]
Since [tex]-21[/tex] is constant with respect to [tex]x[/tex] , the derivative of [tex]-21[/tex] with respect to [tex]x[/tex] is 0 .
Now we have
[tex]4x^{39}y'+156x^{38}y+9y^8y'-52x=0[/tex]
Finally lets solve for [tex]y'[/tex].
Subtract [tex]156x^{38}y[/tex] from both sides of the equation.
[tex]4x^{39}y'+9y^8y'-52x=-156x^{38}y[/tex]
Add [tex]52x[/tex] to both sides of the equation.
[tex]4x^{39}y'+9y^8y'=-156x^{38}y+52x[/tex]
Factor [tex]y'[/tex] out of [tex]4x^{39}y'[/tex]
[tex]y'\left(4x^{39}\right) +9y^8y'=-156x^{38}y+52x[/tex]
Factor [tex]y'[/tex] out of [tex]9y^8y'[/tex]
[tex]y'\left(4x^{39}\right)+y'\left(9y^8\right)=-156x^{38}y+52x[/tex]
Factor [tex]y'[/tex] out of [tex]y'\left(4x^{39}\right)+y'\left(9y^8\right)[/tex]
[tex]y'\left(4x^{39}+9y^8\right)+=-156x^{38}y+52x[/tex]
Divide each term by [tex]4x^{39}+9y^8[/tex]
[tex]\dfrac{y'\left(4x^{39}+9y^8\right)}{4x^{39}+9y^8} =\dfrac{-156x^{38}y}{4x^{39}+9y^8} +\dfrac{52x}{4x^{39}+9y^8}[/tex]
Cancel the common factor of [tex]4x^{39}+9y^8[/tex] on the left side of the equation.
[tex]y'=\dfrac{-156x^{38}y}{4x^{39}+9y^8} +\dfrac{52x}{4x^{39}+9y^8}[/tex]
Combine the numerators over the common denominator.
[tex]y'=\dfrac{-156x^{38}y+52x}{4x^{39}+9y^8}[/tex]
Factor [tex]52x[/tex] out of [tex]-156x^{38}y+52x[/tex] and simplify.
[tex]y'=\dfrac{52x\left(-3x^{37}y\right)+52x}{4x^{39}+9y^8}[/tex]
[tex]y'=\dfrac{52x\left(-3x^{37}y\right)+52x(1)}{4x^{39}+9y^8}[/tex]
[tex]y'=\dfrac{52x\left(-3x^{37}y+1\right) }{4x^{39}+9y^8}[/tex]
[tex]y'=-\dfrac{52x\left(3x^{37}y-1\right) }{4x^{39}+9y^8}[/tex]
Rewrite [tex]y'[/tex] as [tex]\dfrac{dy}{dx}[/tex].
[tex]\dfrac{dy}{dx}=-\dfrac{52x\left(3x^{37}y-1\right) }{4x^{39}+9y^8}[/tex]
Learn more about the Power Rule for derivatives here
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Learn more about the Sum Rule for derivatives here
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Learn more about the Chain Rule for derivatives here
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At which point do the lines y=4x+1 and y=2x-1 intersect
The lines having linear equation y=4x+1 and y=2x-1 intersect at the point (-1, -3).
What is a linear equation, exactly?
A linear equation is an algebraic equation of the first degree that describes a line in a two-dimensional plane. It is an equation of the form y = mx + b, where x and y are variables, m is the slope of the line, and b is the y-intercept. The slope represents the rate of change of y with respect to x, while the y-intercept represents the point at which the line crosses the y-axis
Now,
To find the point of intersection between the lines y=4x+1 and y=2x-1, we can set the two equations equal to each other or we can graph these and find the intersection point.
1.
4x+1 = 2x-1
Simplifying this equation, we get:
2x = -2
x = -1
Now, we can substitute this value of x into either equation to find the corresponding value of y:
y = 4x+1 = 4(-1)+1 = -3
Therefore,
the lines y=4x+1 and y=2x-1 intersect at the point (-1, -3).
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A boy owes his grandmother $1000. She paid for his first semester at community college. The conditions of the loan are that he must pay her back the whole amount in one payment using a simple interest rate of %7 per year. She doesn't care in which year he pays her. The table contains input and output values to represent how much money he will owe his grandmother 1, 2, 3, or 4 years from now.
Year --------Amount owed (in $)
1 -------1070
2 ------------1140
3 -----------1210
4 ----------1280
A.
Yes, because the rate of change is a constant $___
nothing per year.
B. No, because the rate of change is $____
nothing per year during the second year and $___
nothing per year during the third year.
A. Yes, because the rate of change is a constant $70 per year. We can observe that the amount owed increases by $70 for every year that passes.
This is due to the simple interest rate of 7% per year, which means that the boy owes an additional 7% of the original amount ($1000) each year. Thus, the the amount owed after one year is $1000 + 0.07($1000) = $1070, after two years it is $1000 + 2(0.07($1000)) = $1140, and so on. $70 is the change rate per year.
B. It will be no due to the change rate is $0 per year in 2nd and 3rd years.
This statement is false because, as we calculated in part A, the rate of change is constant at $70 per year. Therefore, the amount owed will increase by $70 even during the second and third years, as long as the boy has not made any payments towards the loan.
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Help plsss
Determine if it’s linear