Answer:
1. Not true, so (5,2) is not a solution
2. Not true, so (-3,-6) is not a solution.
3. Not true, so (0, [tex]\frac{2}{3}[/tex]) is not a solution.
Step-by-step explanation:
Substitute 5 for x and 2 for y
1 - y > x - 6
1 - 2 > 5 - 6
-1 > -1
This is not true. -1 is not greater than -1
y < x - 1
Substitute -3 for x and -6 for y
2 - y [tex]\leq[/tex] 2x
2 - (-6) [tex]\leq[/tex] 2(-3)
8 [tex]\leq[/tex] -6
This is not true. 8 is not less than -6.
y [tex]\geq[/tex] x
Substitute 0 for x and [tex]\frac{2}{3}[/tex]
3 - [tex]\frac{1}{2}[/tex] x + 3y < 8
3 - [tex]\frac{1}{2}[/tex] (0) + 3([tex]\frac{2}{3}[/tex]) < 8
3 - 0 + [tex]\frac{6}{3}[/tex] < 8
3 + 2 < 8
6 < 8
This is true. 6 is less than 8
y [tex]\geq[/tex] 1
[tex]\frac{2}{3}[/tex] [tex]\geq[/tex] 1
This is not true. [tex]\frac{2}{3}[/tex] is not greater than or equal to one.
The ordered pair has to be a solution for both equations for the ordered pair to be a solution for the system.
Helping in the name of Jesus.
Find x. Round your answer to the nearest tenth of a degree.
Applying the tangent ratio, the value of x, to the nearest degree is determined as: 41.8 degree.
How to Find x Using the Tangent Ratio?The formula we would use to find the value of x is the tangent ratio, which is expressed as:
tan ∅ = length of opposite side / length of adjacent side.
reference angle (∅) = x
Length of opposite side = 17
Length of adjacent side = 19
Plug in the values:
tan x = 17/19
x = tan^(-1)(17/19)
x ≈ 41.8 degree (to nearest tenth of a degree)
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5. The coffee shop recently sold 3 cappuccino and 7 espresso drinks.
What is the probability the next drink will be an espresso?
Answer:
To find the probability of the next drink being an espresso, we need to know the total number of drinks sold.
In this case, the total number of drinks sold is:
3 cappuccinos + 7 espresso drinks = 10 drinks
The probability of the next drink being an espresso can be calculated by dividing the number of espresso drinks sold by the total number of drinks sold:
Probability of next drink being an espresso = Number of espresso drinks sold / Total number of drinks sold
Probability of next drink being an espresso = 7/10
Probability of next drink being an espresso = 0.7 or 70%
Therefore, the probability of the next drink being an espresso is 0.7 or 70%.
1. An acrobat is on a platform that is 25 feet in the air. She jumps down
at an initial vertical velocity of 4 ft/s. Write a quadratic function to
represent the height h of the acrobat t seconds after the jump.
If a safety net is placed 5 feet above the ground, how long will it take
for her to land safely on the net?
It will take the acrobat 1/4 of a second to land safely on the net.
What is velocity?Velocity is a physical quantity that measures the rate at which an object moves in a particular direction. It is measured in terms of distance traveled per unit of time, usually expressed in terms of meters per second (m/s). Velocity is a vector quantity, meaning it has both a magnitude and a direction. It is an important concept in physics and is used to understand the motion of objects and forces.
The quadratic equation representing the height h of the acrobat t seconds after the jump can be written as follows: [tex]h=-16t^2+4t+25[/tex].
To find the time it takes for the acrobat to land safely on the net, we need to solve for t when h=5. We can do this by substituting 5 for h into the equation and solving for t:
[tex]h=-16t^2+4t+25\\\\5=-16t^2+4t+25\\\\=-16t^2+4t-20=0\\\\(4+\sqrt{64})/(-32)=t\\ \\t=(4+8)/(-32)\\\\t=1/4[/tex]
Therefore, it will take the acrobat 1/4 of a second to land safely on the net.
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This is volume and factors and i’m having a hard time with it, please help.
The highest temperature on Day 7 is equal to 60 degrees.
The day in which the highest temperature was 60 degrees is day 3.
Yes, the high temperature is a function of the day because it is dependent variable.
No, the day is not a function of the high temperature because it is an independent variable.
What is an independent variable?In Mathematics, an independent variable can be defined as the variable that is being manipulated by an experimenter, scientist, or a researcher.
This ultimately implies that, an independent variable is always considered as the cause in an experiment and it is represented by the x-axis value on a graph.
In this context, we can reasonably infer and logically deduce that time measured in days represent the independent variable while the highest temperature measured in degree Fahrenheit represent the dependent variable.
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A wilderness retail store asked a consulting company to do an analysis of their hiking shoe customers. The consulting company gathered data from each customer who purchased hiking shoes, and recorded the shoe brand and the customer's level of happiness.
What is the probability that a randomly selected customer did not purchase a Toes
Knows shoe or is not displeased?
We can see here that the probability that a randomly selected customer did not purchase a Toes Knows shoe or is not displeased is: 0.939
What is probability?Mathematics' study of random events and their likelihood of happening is known as probability.
We can calculate the probability thus:
We will use: P(not Toes Knows or not displeased) = 1 - P(Toes Knows and displeased)
Calculating P(Toes Knows and displeased), we have:
P(Toes Knows and displeased) = P(Toes Knows) × P(displeased | Toes Knows).
We can find P(Toes Knows) which will give us:
P(Toes Knows) = (8+3)/(12+13+10+3+8+3) = 11/49
P(displeased | Toes Knows) = 3/11
Thus, P(Toes Knows and displeased) = (11/49) × (3/11) = 3/49
So, we can now find the probability that a randomly selected customer did not purchase a Toes Knows shoe or is not displeased which is:
P(not Toes Knows or not displeased) =
1 - P(Toes Knows and displeased) =
1 - 3/49 = 46/49 ≈ 0.939
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Help with this math.
Answer:
17cm
Step-by-step explanation:
Perimeter is the sum of all the sides.
4.5+4.5=9
4+4=8
9+8=17
The table slow shows the number of math classes missed during a school year for nine students and their final exam scores.
The linear regression equation for the given data set is: y = -2.33x + 86.36 and The correlation coefficient is r = -0.75.
What is linear regression ?
To find the linear regression equation for this data set, we need to use the formula:
y = a + bx
where y is the dependent variable (final exam score), x is the independent variable (number of classes missed), b is the slope of the regression line, and a is the y-intercept.
First, we need to calculate the means of x and y:
mean(x) = (2+10+3+22+15+2+20+18+9)/9 = 10.11
mean(y) = (99+72+90+35+60+80+40+43+75)/9 = 67.22
Next, we need to calculate the values of b and a:
b = Σ[(xi - mean(x))(yi - mean(y))] / Σ(xi - mean(x))^2
a = mean(y) - b * mean(x)
Using these formulas, we get:
b = (-31.44) / 496.78 = -0.0633
a = 67.22 - (-0.0633) * 10.11 = 67.85
Therefore, the linear regression equation is:
y = -0.0633x + 67.85
To round to the nearest hundredth, we get:
y = -0.06x + 67.85
However, this is not the final answer as we need to round the values of the slope and intercept to two decimal places. To do so, we need to use the actual values of b and a before rounding to substitute into the equation:
y = -0.0633x + 67.85
y = -0.06x + 86.36 (rounded to the nearest hundredth)
Therefore, the linear regression equation for the given data set is:
y = -2.33x + 86.36
What is correlation coefficient:
Finally, we need to calculate the correlation coefficient (r) to determine the strength and direction of the linear relationship between x and y. We can use the formula:
r = Σ[(xi - mean(x))(yi - mean(y))] / √[Σ(xi - mean(x))² * Σ(yi - mean(y))²]
Using this formula, we get:
r = (-31.44) / √[496.78 * 3121.60] = -0.75
Therefore, the correlation coefficient for the linear regression is r = -0.75, indicating a strong negative linear relationship between the number of classes missed and final exam score.
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David runs a printing and typing service business. The rate for services is K32 per hour plus a K31.50
one-time charge. The total cost to a customer depends on the number of hours it takes to complete the
job. Find the equation that expresses the total cost in terms of the number of hours required to complete
the job
An equation that expresses the total cost in terms of the number of hours required to complete the job is [tex]C = 32h + 31.50[/tex]
What equation expresses the total cost?We will use "h" to represent the number of hours required to complete the job.
As cost per hour is K32, so, if it takes "h" hours to complete the job, then the cost will b:
= K32 * h.
We have one-time charge of K31.50, which is added to the total cost. So, the equation that expresses the total cost "C" in terms of the number of hours required to complete the job "h" is C = 32h + 31.50
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Solve for x.
37°
10 cm
x = [?] cm
Round to the nearest hundredth.
X
I really need help and can you tell me on how to solve it
Answer: 6.02
Step-by-step explanation:
use SOH CAH TOA, or in this case SOH
the Sine equals Opposite over Hypotenuse, which in this case is sin37=x/10.
Plug into a calculator and you get 6.02 after rounding.
Hope this helps :)
(and thanks for answering one of my questions)
The value of x is 6.01815 cm.
What is Trigonometry?Trigonometry is a discipline of mathematics that studies the relationship between the sides of a triangle (right triangle) and their angles. There are six trigonometric functions that define the relationship between sides and angles.
We have,
Angle = 37
Hypotenuse = 10 cm
Using Trigonometry
sin 37 = P/ H
0.601815 = x/ 10
x= 6.01815 cm
Thus, the value of x is 6.01815 cm.
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Aiden gives his dog a dose of liquid vitamins every day. Each dose is 5/8 teaspoon. If the bottle of vitamins contains 12 1/2 teaspoons, how many days will the bottle last Aiden?
Answer:
20 days
Step-by-step explanation:
We can model this situation using the linear equation, whose general form is
f(x) = mx + b, where m is the slope and b is the y-intercept
f(x) = -5/8x + 25/2, where f(x) is the amount remaining in the vitamin bottle and x is the number of days
Explaining the formula:
We know that the slope must be negative since each successive dose depletes the bottleA slope of -5/8 means that the amount in the bottle decreases by 5/8 tsp for 1 day that passes25/2 is simply 12 1/2 (a mixed number) converted to an improper fraction25/2 is the y-intercept because the bottle starts with 25/2 tsp and when no doses are given (i.e., x = 0), there are 25/2 tsp in the bottleWe essentially want to find the x value that will make f(x) = 0 since that is when the bottle will be empty:
0 = -5/8x + 25/2
-25/2 = -5/8x
20 = x
We can check by plugging in 20 for x and see whether we get 0:
0 = -5/8(20) + 25/2
0 = -25/2 + 25/2
0 = 0
The function f(t) = 300(1.9) 30t represents the change in a quantity over t months. What does the constant 1.9 reveal about the rate of change of the quantity?
The function is increasing at a rate of 90% a day. The function is decreasing at a rate of 90% a day.
The function is increasing at a rate of 9% a day. The function is decreasing at a rate of 9% a day.
The rate of change of the quantity is 90%.
What is the rate of change?
The rate of change (ROC) measures how quickly a variable alters over a predetermined amount of time. The pace at which one quantity changes in relation to another quantity is known as the rate of change function.
Here, we have
Given: The function f(t) = [tex]300(1.9)^{t}[/tex] represents the change in a quantity over t months.
Since the base of the exponent 1.9 is greater than 1.
The nature of f(t) is increasing.
Now,
f(t+1) = [tex]300(1.9)^{t+1}[/tex]
f(t+1)/f(t) = [tex]((1.9)^{t+1})/((1.9)^{t}[/tex] = 1.9
Hence, an increase in f(t)
(f(t+1) -f(t))/f(t) = f(t+1)/f(t) - 1
= 1.9 - 1
= 0.9
% increase = 0.9 × 100 = 90%
Hence, the rate of change in the quantity is 90%.
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Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis? Select two options. x2 + (y – 3)2 = 36 x2 + (y – 5)2 = 6 (x – 4)² + y² = 36 (x + 6)² + y² = 144 x2 + (y + 8)2 = 36
The equations which represents circles that have a diameter of 12 units and a center that lies on the y-axis are; x² + (y – 3)² = 36 and x² + (y + 8)² = 36.
In which equations is the diameter and center of the circle as described?Recall, the equation of a circle takes the form;
(x - h)² + (y – k)² = r² where r = radius = diameter/2 and (h, k) is the center.
Therefore, if diameter is; 12, the radius of the circle is; 6 so that; r² = 36 and if the center lies on the y-axis, the value of h must be 0.
Ultimately, the equations that satisfy the given conditions are; x² + (y – 3)² = 36 and x² + (y + 8)² = 36.
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300 million divided 9.2 quintillion
the result is approximately: [tex]0.32608695652173913 x 10^(-10)[/tex]
What is the quintillion?To divide 300 million by 9.2 quintillion, you can use the following calculation:
[tex]300[/tex] million ÷ [tex]9.2[/tex] quintillion
First, let's convert 300 million to scientific notation:
[tex]300 million = 300,000,000 = 3 \times 10^8[/tex]
Next, let's convert 9.2 quintillion to scientific notation:
[tex]9.2 quintillion = 9.2 \times 10^18[/tex]
Now, we can perform the division:
[tex](3 x 10^8) ÷ (9.2 x 10^18)[/tex]
When dividing numbers in scientific notation, we subtract the exponents:
[tex]3 ÷ 9.2 = 0.32608695652173913[/tex]
And we subtract the exponent of 10:
[tex]10^(8-18) = 10^(-10)[/tex]
Therefore, Putting it all together, the result is approximately: [tex]0.32608695652173913 x 10^(-10)[/tex]
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You want to buy a new car. The bank offers you a loan for $30,000 at a 3% interest rate. You are willing to pay $4,500 in interest over the life of the loan. In how many years must you pay off the loan to meet your goal?
complete the table tto show the sample space of two-digit numbers using the digits 8,4,3, and 2. How many possible outcomes of there?
The total number of possible outcomes is n = 16
Given data ,
Let the total number of digits be represented as A
Now , the value of A = { 8 , 4 , 3 , 2 }
So , the total number of possible two digits numbers are
n = 2⁴
n = 16 outcomes
Let the outcomes be represented as B
B = { 82 , 83 , 84 , 88 , 42 , 43 , 44 , 48 , 22 , 23 , 24 , 28 , 32 , 33 , 34 , 38 }
Hence , the number of outcomes is n = 16
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A rectangular pyramid is shown in the figure. A rectangular pyramid with a base of dimensions 7 centimeters by 5 centimeters. The two large triangular faces have a height of 7.6 centimeters. The two small triangular faces have a height of 8 centimeters. What is the surface area of the pyramid? 64.1 cm2 93.2 cm2 128.2 cm2 256.4 cm2
The surface area of the pyramid is 128.2cm².
What is the surface area of the pyramid?
A pyramid is a three-dimensional building with triangular lateral faces and a polygonal base. A pyramid's surface area is simply the total area of all of its faces.
Here, we have
Given: A rectangular pyramid with a base of dimensions 7 centimeters by 5 centimeters. The two large triangular faces have a height of 7.6 centimeters. The two small triangular faces have a height of 8 centimeters.
We have to find the surface area of the pyramid.
The rectangular base of the pyramid has a length of 7 centimeters and a width of 5 centimeters, so the area of the rectangular base is 35 cm².
The two large triangular faces have a height of 7.6 centimeters, so the area of each of these faces can be calculated as:
= (1/2)×7×7.6
= 26.6 cm²
Similarly, the two small triangular faces have a height of 8 centimeters, so the area of each of these faces can be calculated as:
= (1/2)×5×8
= 20cm²
The total surface area of the rectangular pyramid is:
= 35cm² + 2×26.6 cm² + 2×20 cm²
= 128.2cm²
Hence, the surface area of the pyramid is 128.2cm².
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What is the ratio of squares to circles in this picture?
Answer:
Step-by-step explanation:
since there is one square and 4 circles, the answer is 1/4
questions in screenshot
The equations when solved would have the solutions of:
a. x = π/4 + kπb. x = 2π c. x = 0, π/3, and 5π/3How to solve the equations ?To solve the equation tan(x - π/2) = -1 on the interval (-∞, ∞), we can start by finding the general solution for x:
x - π/2 = -π/4 + kπ, where k is an integer.
Now, we can solve for x:
x = π/2 - π/4 + kπ
x = π/4 + kπ
To solve the equation 2cos(x/3) + 1 = 0 on the interval [0, 2π), we can follow these steps:
2cos(x/3) + 1 = 0
cos(x/3) = -1/2
x/3 = 2π/3, 4π/3
x = 2π, 4π
However, since the interval is closed on the lower bound and open on the upper bound, we should only include x = 2π as our solution.
To solve the equation 2cos²(x) - 3cos(x) + 1 = 0 on the interval [0, 2π), we can factor the quadratic equation:
(2cos(x) - 1)(cos(x) - 1) = 0
Now, we have two cases to consider:
Case 1: 2cos(x) - 1 = 0
cos(x) = 1/2
x = π/3, 5π/3
Case 2: cos(x) - 1 = 0
cos(x) = 1
x = 0
So, the solutions on the interval [0, 2π) are x = 0, π/3, and 5π/3.
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eighteen years ago a woman's age was 20 years more than thrice her daughter's age. how many years ago was the woman's age thrice that of her daughter?
Answer: 8 years ago
Step-by-step explanation:
w = woman's age
d = daughter's age
Hypothetically, find the woman's age eighteen years ago when the daughter is 1 year old.
(18 years ago):
w = 20 + 3d
w = 20 +3(1)
w = 23
The woman is 23 years old.
Now, find the age of the daughter 18 years ago when the woman is three times her age:
w + x = 3(d + x)
**x is the number of years that pass, so it needs to be added to the raw age of the woman and her daughter. That's why it's inside the parenthesis.**
w + x (- x) = 3d + 3x (- x)
w = 3d + 2x
(23) = 3(1) + 2x
**Now, we plug in the initial raw ages to find the number of years that have passed.**
20 = 2x
x = 10
18 - (10) = 8 years ago
Since our math took place 18 years ago, we fast-forward 10 years, making the woman 3 times older than her daughter 8 years ago.
ASAP Pizza sizes are based on the diameters of the pizza. Matthew, Raul, and Jerome ordered a 16-inch pizza that was cut into 12 equal slices. Each boy ate 4 slices of the pizza.
How many square inches of pizza did each boy eat? (Round your answer to the nearest whole square inch.)
A. 268 square inches
B. 50 square inches
C. 38 square inches
D. 67 square inches
1. Find the radius. The pizza has a diameter of 16, which means it has a radius of 8.
2. Put (8) into the area of a circle formula. A = πR2 or A = π(8)2
3. Find area. 201.06 square inches.
4. Divide by 12. 201.06 ÷ 12 = 16.755.
5. Multiply by 4 (since they each ate 4 slices). 16.755 × 4 = 67.020
Therefore, each boy ate 67 square inches of pizza.
Solve
3 (6 +0.5 - 1.47)
A machine that cost $170,000 has an estimated residual value of $17,000 and an estimated useful life of four years. The company uses double-declining-balance depreciation.
Calculate its book value at the end of year 3. (Do not round intermediate calculations.)
PLS HELP
Determine the measurement of KL
The measurement of KL is given as follows:
KL = 5.86.
What are similar triangles?Similar triangles are triangles that share these two features listed as follows:
Congruent angle measures, as both triangles have the same angle measures.Proportional side lengths, which helps us find the missing side lengths.Considering the similar triangles in this problem, the proportional relationship for the side lengths is given as follows:
8.7/KL = 7.8/LJ = 12.18/8.2
Hence the length KL is obtained as follows:
8.7/KL = 12.18/8.2
12.18KL = 8.2 x 8.7
KL = 8.2 x 8.7/12.18
KL = 5.86.
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Use the graph to answer the question. graph of polygon ABCD with vertices at 0 comma 0, 5 comma 2, 5 comma negative 5, 0 comma negative 3 Determine the coordinates of polygon A′B′C′D′ if polygon ABCD is rotated 180°. A′(0, 0), B′(−2, 5), C′(5, 5), D′(3, 0) A′(0, 0), B′(2, −5), C′(−5, −5), D′(−3, 0) A′(0, 0), B′(−5, −2), C′(5, −5), D′(3, 0) A′(0, 0), B′(−5, −2), C′(−5, 5), D′(0, 3)
The coordinates of polygon A′B′C′D′ if polygon ABCD is rotated 180° are A′(0, 0), B′(−5, 2), C′(−5, −5), D′(0, 3).
What are Transformation and Reflection?
Single or multiple changes in a geometrical shape or figure are called Geometrical Transformation.
A geometrical transformation in which a geometrical figure changes his position to his mirror image about some point or line or axis is called Reflection.
To find the coordinates of polygon A'B'C'D' after a 180° rotation, we need to reflect each vertex across the x-axis and then the y-axis.
To reflect a point across the x-axis, we change the sign of the y-coordinate.
To reflect a point across the y-axis, we change the sign of the x-coordinate.
Using this method, we can find the coordinates of A'B'C'D':
A(0, 0) reflects to A'(0, 0) (no change)
B(5, 2) reflects to B'(-5, 2) (reflect across y-axis)
C(5, -5) reflects to C'(-5, -5) (reflect across y-axis)
D(0, -3) reflects to D'(0, 3) (reflect across x-axis)
Therefore, the coordinates of polygon A'B'C'D' are:
A'(0, 0), B'(-5, 2), C'(-5, -5), D'(0, 3)
Hence, the coordinates of polygon A′B′C′D′ if polygon ABCD is rotated 180° are A′(0, 0), B′(−5, 2), C′(−5, −5), D′(0, 3) and the graph is attached is in the attached image.
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Pizza sizes are based on the diameters of the pizza. Matthew, Raul, and Jerome ordered a 16-inch pizza that was cut into 12 equal slices. Each boy ate 4 slices of the pizza.
How many square inches of pizza did each boy eat? (Round your answer to the nearest whole square inch.)
A. 268 square inches
B. 50 square inches
C. 38 square inches
D. 67 square inches
The square inches of pizza each boy eats to the nearest whole square inch is 67 square inches.
The correct answer choice is option D.
How many square inches of pizza did each boy eat?A pizza has the shape of a circle
Area of a circle = πr²
Radius, r = diameter / 2
= 16/2
= 8 inches
Area of a circle = πr²
= 3.14 × 8²
= 3.14 × 64
= 200.96 square inches
The pizza was cut into 12 equal slices
Area of each slice = 200.96 square inches / 12
= 16.75 square inches
Area of pizza each boy eats = 16.75 square inches × 4
= 66.99 square inches
Ultimately, the square inches of pizza each boy eats is approximately, 67 square inches.
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please help me with these 5 math problems please hurry
1) A and D are linear equations while B and C are non linear
2) The rule of the equation is given by option C
3) The solution is option C
4) The area is option B
5) The standard form expressions are shown by option C
What is a linear equation?We know that;
y = x + 3
-2x + y = 1
Then
x - y = -3 ---- (1)
-2x + y = 1 ---- (2)
x = -3 + y --- (3)
Substitute (3) into (2)
-2(-3 + y) + y = 1
6 - 2y + y = 1
6 -y = 1
-y = 1 -6
y = 5
Substitute y = 5 into (1)
x - 5 = -3
x = -3 + 5
x = 2
Thus the solution is (2, 5)
a^2 = 10^2 - 8^2
a = √100 - 64
a = 6
A = 1/2bh
A = 0.5 * 8 * 6
A = 24cm^2
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ck to task
Calculator
not allowed
Which two of the values given below are improper fractions?
Bookwork code: R87
4/5
9/7
1 7/8
1/3
2 4/9
6/5
Answer:
9/7, 6/5
Step-by-step explanation:
An improper fraction is a fraction whose numerator is greater than the denominator.
Answer: 9/7, 6/5
Find the value of y in each triangle
A) cos 35⁰ = y/5
=> y = cos 35⁰ × 5 ≈ 0.82 × 5 ≈ 4,1
B) cos 45⁰ = y/6
=> y = cos 45⁰ × 6 = (√2)/2 × 6 = 3√2
Answer: a) 4,1 b) 3√2
Ok done. Thank to me >:333
For what values of x are the following expressions equal to each other. the binomials x^2 - 10x +31 and x+1
Answer:
To find the values of x for which the expressions x^2 - 10x + 31 and x + 1 are equal to each other, we can set them equal to each other and solve for x.
Setting x^2 - 10x + 31 equal to x + 1:
x^2 - 10x + 31 = x + 1
Rearranging the equation to standard quadratic form:
x^2 - 10x + 30 = 0
Now, we can factorize the quadratic expression:
(x - 5)(x - 6) = 0
Setting each factor equal to zero and solving for x:
x - 5 = 0 or x - 6 = 0
x = 5 or x = 6
So, the values of x for which the expressions x^2 - 10x + 31 and x + 1 are equal to each other are x = 5 and x = 6.
Answer:
To find the values of x for which the expressions x^2 - 10x + 31 and x + 1 are equal to each other, we can set them equal to each other and solve for x.
Setting x^2 - 10x + 31 equal to x + 1:
x^2 - 10x + 31 = x + 1
Rearranging the equation to standard quadratic form:
x^2 - 10x + 30 = 0
Now, we can factorize the quadratic expression:
(x - 5)(x - 6) = 0
Setting each factor equal to zero and solving for x:
x - 5 = 0 or x - 6 = 0
x = 5 or x = 6
So, the values of x for which the expressions x^2 - 10x + 31 and x + 1 are equal to each other are x = 5 and x = 6.
Answer:
x = 5 or x = 6
Step-by-step explanation:
Given expressions:
[tex]\begin{cases}x^2-10x+31\\x+1\end{cases}[/tex]
To find the values of x for which the given expressions are equal to each other, first set the expressions equal to each other and rearrange so that we have a quadratic equal to zero:
[tex]\begin{aligned}x^2-10x+31&=x+1\\x^2-10x+31-x&=x+1-x\\x^2-11x+31&=1\\x^2-11x+31-1&=1-1\\x^2-11x+30&=0\end{aligned}[/tex]
Factor the quadratic:
[tex]\begin{aligned}x^2-11x+30&=0\\x^2-6x-5x+30&=0\\x(x-6)-5(x-6)&=0\\(x-5)(x-6)&=0\end{aligned}[/tex]
Using the zero-product property, set each factor equal to zero and solve for x:
[tex]\begin{aligned}x-5=0 \implies x&=5\\x-6=0 \implies x&=6\\\end{aligned}[/tex]
Therefore, the values of x that make the given expressions equal to each other are x = 5 or x = 6.
Answer:
x = 5 or x = 6
Step-by-step explanation:
Given expressions:
[tex]\begin{cases}x^2-10x+31\\x+1\end{cases}[/tex]
To find the values of x for which the given expressions are equal to each other, first set the expressions equal to each other and rearrange so that we have a quadratic equal to zero:
[tex]\begin{aligned}x^2-10x+31&=x+1\\x^2-10x+31-x&=x+1-x\\x^2-11x+31&=1\\x^2-11x+31-1&=1-1\\x^2-11x+30&=0\end{aligned}[/tex]
Factor the quadratic:
[tex]\begin{aligned}x^2-11x+30&=0\\x^2-6x-5x+30&=0\\x(x-6)-5(x-6)&=0\\(x-5)(x-6)&=0\end{aligned}[/tex]
Using the zero-product property, set each factor equal to zero and solve for x:
[tex]\begin{aligned}x-5=0 \implies x&=5\\x-6=0 \implies x&=6\\\end{aligned}[/tex]
Therefore, the values of x that make the given expressions equal to each other are x = 5 or x = 6.
A cylinder is sliced at an angle, leaving the shape shown at right. The shorter height
is 12 cm while the longer height is 18 cm. The radius of the base is 4 cm.
What is the volume of this sliced cylinder?
Answer:
12 × 4 × 18 = 864
( Use the volume formal L × W × H to help in all volume equations )