Answers
The measure of angle A is 45°.
The measure of side c is 8.5'.
Given a right angled triangle.
The measure of two sides are given.
The third side can be found using the Pythagoras Theorem.
Pythagoras theorem states that for a right angled triangle, the square of the hypotenuse is the sum of the squares of base and altitude.
Using this,
c² = 6² + 6²
c² = 72
c = √72 = 8.485 ≈ 8.5'
The sides opposite two angles other than the right angle in the triangle are equal.
This means that the angles opposite equal sides will be equal.
So sum of the two angles will be 180 - 90 = 90°
Since they are equal,
∠A = 45°
Hence A = 45° and c = 8.5'
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Related Questions
Compute the determinants using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.
[ 0 4 1
5 −3 0
2 3 1 ]
Answers
Answer:
The determinant is 1Step-by-step explanation:
Given the 3* 3 matrices \(\left[\begin{array}{ccc}0&4&1\\5&-3&0\\2&3&1\end{array}\right]\), to compute the determinant using the first row means using the row values [0 4 1 ] to compute the determinant. Note that the signs on the values on the first row are +0, -4 and +1
Calculating the determinant;
\(= +0\left[\begin{array}{cc}-3&0\\3&1\\\end{array}\right] -4\left[\begin{array}{cc}5&0\\2&1\\\end{array}\right] +1\left[\begin{array}{cc}5&-3\\2&3\\\end{array}\right] \\\\= 0 - 4[5(1)-2(0)] +1[5(3)-2(-3)]\\= 0 -4[5-0]+1[15+6]\\= 0-20+21\\= 1\)
The determinant is 1 using the first row as co-factor
Similarly, using the second column \(\left[\begin{array}{c}4\\-3\\3\end{array}\right]\) as the cofactor, the determinant will be expressed as shown;
Note that the signs on the values are -4, +(-3) and -3.
Calculating the determinant;
\(= -4\left[\begin{array}{cc}5&0\\2&1\\\end{array}\right] -3\left[\begin{array}{cc}0&1\\2&1\\\end{array}\right] -3\left[\begin{array}{cc}0&1\\5&0\\\end{array}\right] \\\\= -4[5(1)-2(0)] - 3[0(1)-2(1)] -3[(0)-5(1)]\\= -4[5-0] -3[0-2]-3[0-5]\\= -20+6+15\\= -20+21\\= 1\)
The determinant is also 1 using the second column as co factor.
It can be concluded that the same value of the determinant will be arrived at no matter the cofactor we choose to use.
Calculate the equivalent ratio 1.25 : 3.75 : 7.5
Answers
The equivalent ratio of 1.25 : 3.75 : 7.5 is 5 : 15 : 30.
To calculate the equivalent ratio of 1.25 : 3.75 : 7.5, we need to find a common multiplier that can be applied to all the numbers in the ratio to make them whole numbers. In this case, the common multiplier is 4 because it can be multiplied to each number to eliminate the decimals.
By multiplying each number in the ratio by 4, we get:
1.25 * 4 = 5
3.75 * 4 = 15
7.5 * 4 = 30
So the equivalent ratio of 1.25 : 3.75 : 7.5 is 5 : 15 : 30.
This means that the relative sizes or quantities represented by the original ratio are maintained in the equivalent ratio. For example, if we had 1.25 units of something, it would be equivalent to 5 units in the new ratio, and if we had 7.5 units, it would be equivalent to 30 units in the new ratio.
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Using slope-intercept, standard form, or point slope -
An arrow released from a bow, traveling towards a target 200 yards away,
reaches a max height of 42.6 ft in only 1.5 seconds. What is the height of the
arrow at 2.5 seconds?
Thank you so much!
Answers
The height of the arrow at 2.5 seconds is 71 feet.
Given:
An arrow released from a bow, traveling towards a target 200 yards away,reaches a max height of 42.6 ft in only 1.5 seconds.
1.5 seconds = 42.6 feet
divide by 1.5 on both sides.
1 sec = 28.4 feet
0.5 sec = 28.4/2
0.5 sec = 14.2 feet
2.5 sec = 1.5 sec + 1 sec
= 42.6 + 28.4
= 71 feet
Therefore The height of the arrow at 2.5 seconds is 71 feet.
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Slope 3/5 yintercept2 write an equation slope-intercept form
Answers
The equation of the line in slope-intercept form is y = (3/5)x + 2. This form allows us to easily identify the slope and y-intercept of the line and to graph it on a coordinate plane.
To write the equation of a line in slope-intercept form, we use the formula:y = mx + b
where:
- y represents the dependent variable (the vertical axis)
- x represents the independent variable (the horizontal axis)
- m represents the slope of the line
- b represents the y-intercept, the point where the line intersects the y-axis.
In this case, we are given the slope as 3/5 and the y-intercept as 2. Plugging these values into the formula, we get:
y = (3/5)x + 2
This equation represents a line with a slope of 3/5, indicating that for every 5 units we move horizontally (along the x-axis), the line moves 3 units vertically (along the y-axis). The y-intercept of 2 tells us that the line intersects the y-axis at the point (0, 2).
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What is the sine of 0?
(Need help)
Answers
The angle of sinθ between the horizontal vector (1, 0) and the slant vector (15/17, -8/17) is sin⁻¹(8/17), which is approximately 29.11 degrees.
To find the angle of sinθ between a horizontal vector and a slant vector, we can use the dot product formula:
a · b = |a| |b| cos(θ)
where a and b are vectors, |a| and |b| are their magnitudes, and theta is the angle between them.
In this case, the horizontal vector is (1, 0) and the slant vector is (15/17, -8/17).
The magnitude of the horizontal vector is 1, and the magnitude of the slant vector is:
|b| = sqrt((15/17)² + (-8/17)²) = sqrt(225/289 + 64/289) = sqrt(289/289) = 1
The dot product of the two vectors is:
a · b = (1)(15/17) + (0)(-8/17) = 15/17
So we have:
15/17 = (1)(1) cos(θ)
cos(θ) = 15/17
To find sin(θ), we can use the trigonometric identity:
sin²(θ) + cos²(θ) = 1
sin²(θ) = 1 - cos²(θ) = 1 - (15/17)² = 64/289
Taking the square root of both sides, we get:
sin(theta) = sqrt(64/289) = 8/17
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Which graph represents 6x-2y>-11
Answers
Answer:
Answer:
C
Step-by-step explanation:
Remember when using the inequality sign < or > the line will be dotted so we can immediately eliminate answer choices b and d
Now let us solve for y
6x-2y>-11
step 1 subtract each side by 6x
now we have
-2y>-11-6x
step 2 divide each side by -2
-11-/2=11/2
-6/-2=3
IMPORTANT -when you divide inequalities by a negative number you flip the inequality sign
So now we have
See image below for reference
When the inequality is facing x the solutions are going to be on top of the line
hope this helps!
Step-by-step explanation:
Hector works 12 hours per week at a part-time job. His original pay rate was $8.00 per hour, but he recently received a raise of x dollars per hour. His new total weekly pay, y, is represented by the equation y=12(8+x)
If Hector earned $126 last week, what was the amount of Hector’s raise per
Answers
Answer:
x=8/3
Step-by-step explanation:
I hope this helps you
44 points in 4 quarters =
Answers
Answer:
11 points per quarter
Step-by-step explanation:
44 / 4 = 11
11
Because 44 divided by 4 = 11
~Roni
pls help i will mark brainlyest
Answers
Answer:
∠b=82°
Step-by-step explanation:
143°=∠c+∠b (exterior angle property)
143°=61°+∠b
∠b=143-61
∠b=82°
Find the area of the figure
Answers
The freezing temperature of the water is 32 degrees Fahrenheit and the boiling temperature of water is 212 degrees Fahrenheit. Write an absolute value equation that represents the minimum and maximum temperature of liquid water. Use X to represent the temperature.
Answers
Answer:
90 = | x - 122 |
Step-by-step explanation:
Given;
maximum temperature, 212 °F
minimum temperature, 32 °F
First, determine the difference of both values;
212 - 32 = 180
Divide this value by 2
180/2 = 90
90 = | x - 212 + 90|
90 = | x - 122 |
where;
x is the temperature of water
Thus, the absolute value equation that represents the minimum and maximum temperature of liquid water is 90 = | x - 122 |
Which situation is modeled here? an item marked down 20% to $12 an item marked down 12% to $9.60 an item marked down 20% to $9.60 an item marked up 20% from $12
Answers
Answer: Your answer is, “an item marked down 20% to $9.60.
Step-by-step explanation:
Answer:
Step-by-step explanation: it is c
A gaming developer company conducted a survey and found that families with teenagers spend, on average $238.92 in playing video games (including downloads, streaming, physical copies, and in-game purchases) a year. Assume the standard deviation is $31.38. Assume the variable is normally distributed. (a) Find the probability that a family spends over $290 in playing video games every year. (b) Find the probability that a family spends less than $150 in playing video games every year. (c) Find the probability that a family spends between $175 and $255 in playing video games every year. (a) The probability that a family spends over $290 in playing video games every year is A . (b) The probability that a family spends less than $150 in playing video games every year is B . (c) The probability that a family spends between $175 and $255 in playing video games every year is C . Enter an answer • 5 points
Answers
Using Standard normal distribution, The probability that a family spends between $175 and $255 in playing video games every year is approximately 0.4821 or 48.21%.
Describe Probability?Probability is a branch of mathematics that deals with the likelihood or chance of an event occurring. It is defined as the measure of the likelihood of an event happening, expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of rolling a 3 on a fair six-sided die is 1/6 because there is only one favorable outcome (rolling a 3) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).
There are two types of probability: theoretical probability and experimental probability. Theoretical probability is based on mathematical analysis and assumes that all possible outcomes are equally likely to occur. Experimental probability, on the other hand, is based on actual observations and measurements of events and their outcomes.
Probability theory has many practical applications in fields such as statistics, economics, engineering, and physics. It can be used to model and predict the behavior of complex systems, such as stock prices, weather patterns, and the spread of diseases. Additionally, probability theory is used in decision-making processes to determine the most favorable course of action based on the likelihood of different outcomes.
(a) To find the probability that a family spends over $290 in playing video games every year, we need to standardize the value using the z-score formula:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
So, z = (290 - 238.92) / 31.38 = 1.63
Using a standard normal distribution table or a calculator, we can find that the probability of a z-score greater than 1.63 is 0.0516.
Therefore, the probability that a family spends over $290 in playing video games every year is approximately 0.0516 or 5.16%.
(b) To find the probability that a family spends less than $150 in playing video games every year, we again need to standardize the value using the z-score formula:
z = (x - μ) / σ
So, z = (150 - 238.92) / 31.38 = -2.83
Using a standard normal distribution table or a calculator, we can find that the probability of a z-score less than -2.83 is 0.0023.
Therefore, the probability that a family spends less than $150 in playing video games every year is approximately 0.0023 or 0.23%.
(c) To find the probability that a family spends between $175 and $255 in playing video games every year, we need to standardize the values of both $175 and $255 using the z-score formula:
z1 = (175 - 238.92) / 31.38 = -2.03
z2 = (255 - 238.92) / 31.38 = 0.51
Using a standard normal distribution table or a calculator, we can find the probability of a z-score between -2.03 and 0.51 is 0.4821.
Therefore, the probability that a family spends between $175 and $255 in playing video games every year is approximately 0.4821 or 48.21%.
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1/2A+N=d
Solve for A
Answers
Answer:
A= 1/2d -1/2N
Step-by-step explanation:
Answer the questions below to find the total surface area of the can.
Answers
Ab=3.14xRadious to the power of 2
then to the area of the rectange you do B x H = YOUR ANSWER
3) State an equation in slope-intercept form that contains the point (–7, 2) and is PARALLEL to the line –x + 3y = 1. Hint: Put the equation in slope-intercept form and recall that the slopes of parallel lines are the same.
Answers
An equation in slope-intercept form that contains the point (-7, 2) and is parallel to the line -x + 3y = 1 is y = (1/3)x + 7/3.
What is the slope-intercept form?
The slope-intercept form is a way of writing the equation of a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
To find an equation of the line parallel to the given line and passing through the point (-7, 2), we need to first find the slope of the given line.
-x + 3y = 1
Add x to both sides:
3y = x + 1
Divide both sides by 3:
y = (1/3)x + 1/3
The slope of the given line is 1/3.
Since the line we want is parallel to this line, it must have the same slope.
So we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the given point.
Plugging in the values we have:
y - 2 = (1/3)(x + 7)
To get the equation in slope-intercept form, we can solve for y:
y = (1/3)x + 7/3
Therefore, an equation in slope-intercept form that contains the point (-7, 2) and is parallel to the line -x + 3y = 1 is y = (1/3)x + 7/3.
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Four students have interviews for summer internships. One is a first-year student, two are second-year students, and one is a third-year student. They are interviewed in the following order: third-year student, second-year student, first-year student, and second-year student. The following conditions apply:
Ashley interviewed after Carlos but before Brianna.
David interviewed before Carlos.
Brianna interviewed last.
Who is the first-year student?
Answers
Answer:
Ashley
Step-by-step explanation:
Lets understand the conditions.
Carlos < Ashley < Brianna
David < Carlos
Clearly, this tells us that David < Carlos < Ashley < Brianna, meaning the order is
David, Carlos, Ashley, Brianna
Since Ashley was interviewed third, she is the first year student
Cheers
there are 36 green, 22 white,30 purple, and 14 blue gumballs in the gumball machine. what is the probability of getting a white gumball
Answers
probability of getting a white gumball
= 22/102
5. John is tiling the living room floor of his home. The living room is a rectangular shape
room that is 10 ft. 9in, by 15ft. 3 in.
a. What is the perimeter of the room?
B. How many 6in by 6in square tiles are required to cover the floor?
C. Calculate 10% of the answer from
Part b (this accounts for the overage due to grout and waste)
Answers
The perimeter of the room is 624 inches.
We need 657 tiles to cover the floor.
We need to add 66 extra tiles to account for the overage due to grout and waste.
What is a rectangle?A rectangle is a 2-D shape with length and width.
The length and width are different.
If the length and width are not different then it is a square.
The area of a rectangle is given as:
Area = Length x width
We have,
Length = 10 ft 9in
Width = 15ft 3 in
a.
Perimeter = 2 (Length + Width)
Length = 10 ft 9 in = (10 x 12) + 9 = 129 in.
Width = 15 ft. 3 in. = (15 x 12) + 3 = 183 in
Perimeter.
= 2(129 + 183)
= 624 in.
b.
Area = Length x Width
Area = 129 in. x 183 in
Area == 23,607 in²
Each tile covers an area of 6 in. x 6 in
Area of tiles = 36 in²
So,
The number of tiles required.
= Area ÷ Tile area
= 23,607 in² ÷ 36 in²
≈ 656.3 tiles.
c.
10% of 657
= 10/100 x 657
= 0.1 x 657
= 65.7
= 66
Thus,
The perimeter of the room is 624 inches.
We need 657 tiles to cover the floor.
We need to add 66 extra tiles to account for the overage due to grout and waste.
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A - on the po boyds at a emase the foot, 1, of building. He. Observes an obje- et on the top, P of the building at an angle of ele- building of 66 Aviation of 66 Hemows directly backwards to new point C and observes the same object at an angle of elevation of 53° · 1P) |MT|= 50m point m Iame horizontal level I, a a
Answers
Answer:
53\(x_{123}\) == 134 cf
Step-by-step explanation:
A - on the po boyds at a emase the foot, 1, of building. He. Observes an obje- et on the top, P of the building at an angle of ele- building of 66 Aviation of 66 Hemows directly backwards to new point C and observes the same object at an angle of elevation of 53° · 1P) |MT|= 50m point m Iame horizontal level I, a a
The height of the building is approximately 78.63 meters.
The following is a step-by-step explanation of how to solve the problem. We'll need to use some trigonometric concepts and formulas to find the solution.
Draw a diagram of the situation described in the problem to get a better understanding of the problem. The diagram would have a right-angled triangle with angle of elevation of 66° at the bottom left vertex and another angle of elevation of 53° at the bottom right vertex. The object on top of the building is at the vertex of the triangle. Point M and I on the diagram are points on the horizontal line of sight and on the ground respectively. We can label the diagram with the following values:Angle of elevation from point A = 66°Angle of elevation from point P = 53° Length of line segment AM = h Length of line segment MP = x Length of line segment IP = y Length of line segment MT = 50m. We'll use these values to calculate the length of h, which is the height of the building.Use the tangent ratio to find x:tan 66° = h / x => x = h / tan 66°. Use the tangent ratio to find y:tan 53° = h / y => y = h / tan 53°.We know that x + y = 50, so substituting the expressions for x and y from step 3 gives:h / tan 66° + h / tan 53° = 50h = 50 tan 66° tan 53° / (tan 53° + tan 66°) ≈ 78.63 m.Therefore, the height of the building is approximately 78.63 meters.
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Jin has a collection of 60 movies. Twenty-four of the movies are animated movies. What percent of Jin’s movie collection are animated movies?
Answers
Answer:
Uh, 2400%, I think? I think my answer might be incorrect
hope this helped :)
Can I get help with this, I’m so confused
Answers
The linear and exponential equations for the amount of money in the banks indicates;
TCF Bank Equation
y = 100·x + 1000
Well Fargo Bank Equation
y = (1.06)ˣ
The number of years it will take for both accounts to have the same amount of money is about 17 years
What is an exponential equation?An exponential equation is an equation of the form; y = a·bˣ, where the input variable, x is an exponent.
The equation for calculating the amount in each bank, based on the details are;
TCF Bank;
Amount invested = $1,000
Amount added each year = 100·x
The TCF Bank Equation is; y = 100·x + 1000Well Fargo Bank Equation
The percentage of the amount earned as interest each year = 6% = 0.06
The exponential equation for compound interest amount is; y = 1000·(1 + 0.06)ˣ
Therefore; y = 1000·(1.06)ˣ
The Well Fargo Bank Equation is; y = 1000·(1.06)ˣWhen the amount in the two accounts have the same amount of money, we get;
1000·(1.06)ˣ = 100·x + 1000
Therefore; (1.06)ˣ = (100·x + 1000)/1000 = 0.1·x + 1
(1.06)ˣ = 0.1·x + 1
Solving the above equation using the substitution method and graphing indicates that we get;
x = 0, and x ≈ 17.125732
Therefore, it takes about 17 years for the two accounts to have the same amount of money
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The ratio of the sides of rectangle LMNP to the sides of rectangle TUVW is 1:4. The length of LM is 3.6 in, and the length of UV is 16 in.
What is the difference between the areas of the two rectangles?
A. 226 in2
B. 172.8 in2
C. 211 in2
D. 216 in2
Answers
what must be multiplied to 95 to make it a perfect square
Answers
Answer:
here is the answer
Step-by-step explanation:
Thus, the square of 95 will be 9025.
Answer:
95 itself------------------------
Find prime factors of 9595 = 5*19To be a perfect square both prime factors must be squares so the least number to be multiplied is:
5*19 = 95What is the Pythagorean theorem
Answers
Answer:
The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. Observe the following triangle ABC, in which we have BC2 = AB2 + AC2. Here, AB is the base, AC is the altitude (height), and BC is the hypotenuse. It is to be noted that the hypotenuse is the longest side of a right-angled triangle.
Pythagoras Theorem Equation
The Pythagoras theorem equation is expressed as, c2 = a2 + b2, where 'c' = hypotenuse of the right triangle and 'a' and 'b' are the other two legs. Hence, any triangle with one angle equal to 90 degrees produces a Pythagoras triangle and the Pythagoras equation can be applied in the triangle.
Answer:
The Pythagoras theorem states that:The sum of the squares on the edges of a right triangle angle is equal to the squareon the hypotenuse.Mathematically stated as: a² +b² = c²Hope this helps.Good luck ✅.
Hot tea is brought to a boil at 212F and must be cooled to a temperature of 160°F The room temperature is 68°F
Newtons law of cooling can be used to find how long it takes a liquid to be cooled. The formula T= (To-Tr)e^-rt +Tr where Tr is the air temperature, To is initial temperature, T is the expected temperature, and r is rate of cooling and t is time in minutes. How long does it take for tea to cool if the rate of cooling is 0.05?
A. 3.9
B. 9.0
C. 12.9
D. 37.9
Answers
Answer:
no explanationm
Step-by-step explanation:
I think it's D
Find \(A(3,4)\).
HINT: \(A(1,n)=2^n\) whenever \(n \geq 1\)
![Find [tex]A(3,4)[/tex].HINT: [tex]A(1,n)=2^n[/tex] whenever [tex]n \geq 1[/tex]](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/gKOa0EWM3CE90odde6WaSqv1p45wZzPW.jpeg)
Answers
Along with proof of (a.) and (d.), (b.) Power tower: one level is a, (k + 1) levels is a raised to the power of a power tower with k levels, (c.) A(2, n) <= 2 ↑↑ n for all positive integers n, where ↑↑ denotes power tower notation.
What is an Ackermann function?The idea of a fully computable function that is not primitive recursive is illustrated by the recursively constructed mathematical function known as the Ackermann function. Since m and n are non-negative integers, it is commonly written as A(m, n).
a.) Prove using regular induction that \(A(1, n) \leq 2^n\) for all positive integers n:
Base Case: For n = 1, A(1, 1) = 2, which is equal to \(2^1\).
Inductive Hypothesis: Assume that \(A(1, k) \leq 2^k\) for some positive integer k.
Inductive Step: We need to show that \(A(1, k + 1) \leq 2^{(k + 1)}\). Using the recursive definition of A(m, n), we have \(A(1, k + 1) = A(0, A(1, k)) = 2^{(A(1, k))}\leq 2^{(2^k)}\) (by inductive hypothesis)\(< = 2^{(2^{(k + 1)})}\).
Therefore, by regular induction, we have proved that \(A(1, n) \leq 2^n\) for all positive integers n.
b.) A power tower with one level is defined as a, and a power tower with (k + 1) levels is defined as a raised to the power of a power tower with k levels.
c.) \(A(2, n) \leq 2\) ↑↑ n for all positive integers n, where ↑↑ denotes power tower notation.
d.) The recursive definition of a triple arrow-up notation for power towers is:
a ↑↑↑ 1 = a (base case)
a ↑↑↑ (k + 1) = a ↑↑ (a ↑↑↑ k) (recursive step)
This definition states that a triple arrow-up notation with one level is equal to the base value "a", and a triple arrow-up notation with (k + 1) levels is equal to "a" raised to the power of a triple arrow-up notation with "k" levels.
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Complete Question: ( Refer to image)
![Find [tex]A(3,4)[/tex].HINT: [tex]A(1,n)=2^n[/tex] whenever [tex]n \geq 1[/tex]](https://i5t5.c14.e2-1.dev/h-images-qa/answers/attachments/H0gaDSJwa5RbRxpz9fmXX3WVlDgz75Up.png)
One of the goats on Diane's farm is sick and has a high fever. Over the course of 4 hours, its
temperature changes by-2.8°F. Diane wants to know the average change in the goat's
temperature per hour.
• Write an expression for the average change in the goat's temperature each hour.
- 2.8
4
Answers
Answer:
-2.8 / 4
Step-by-step explanation:
The equivalent expression for the change in the goat's temperature each hour is A = -0.7°F/hour
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let's say the initial temperature of the goat was T1, and the final temperature after 4 hours was T2.
Now , the total change in temperature over the 4 hours is:
T2 - T1 = -2.8°F
The average change in temperature per hour can be expressed as:
The average change in temperature per hour = total change in temperature / number of hours
The average change in temperature per hour = (-2.8°F) / 4 hours
On simplifying the equation , we get
The average change in temperature per hour = -0.7°F/hour
Hence , the expression for the average change in the goat's temperature each hour is A = -0.7°F/hour
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Brainliest if its correct :)
Answers
How do I know if it’s true or false for number 1?
Answers
We want to know if the following statements are true or false
item a)
\(1-\cos x=\sin x\)To solve this one, we're going to use the following identity
\(\sin ^2x+\cos ^2x=1\)Rewriting this expression "isolating" the sine, we have
\(\begin{gathered} \sin ^2x+\cos ^2x=1 \\ \sin ^2x=1-\cos ^2x \\ \sin x=\pm\sqrt[]{1-\cos^2x} \end{gathered}\)Using this in our expression, we have
\(\begin{gathered} 1-\cos x=\sin x \\ \Rightarrow1-\cos x=\sqrt[]{1-\cos^2x} \end{gathered}\)When we have a difference of two squares, we can separate them as
\(a^2-b^2=(a+b)(a-b)\)Using this in the argument of the square root, we have
\(1-\cos ^2x=(1+\cos x)(1-\cos x)\)Using this to rewrite our expression
\(\begin{gathered} 1-\cos x=\sqrt[]{1-\cos^2x} \\ 1-\cos x=\sqrt[]{(1+\cos x)(1-\cos x)} \\ 1-\cos x=\sqrt[]{(1+\cos x)}\sqrt[]{(1-\cos x)} \\ \frac{1-\cos x}{\sqrt[]{1-\cos x}}=\sqrt[]{1+\cos x} \end{gathered}\)We can rewrite the numerator of the left side of the equation as the product of its square root.
\(1-\cos x=(\sqrt[]{1-\cos x})^2\)Using this in our expression, we have
\(\begin{gathered} \frac{(\sqrt[]{1-\cos x})^2}{\sqrt[]{1-\cos x}}=\sqrt[]{1+\cos x} \\ \sqrt[]{1-\cos x}=\sqrt[]{1+\cos x} \\ 1-\cos x=1+\cos x \\ -\cos x=\cos x \end{gathered}\)Since the cosine of an angle is not equal to minus its value, the first statemente is FALSE.
item b)
\(1-\cos ^2x=\sin ^2x\)We're going to use the same identity to solve this one.
\(\sin ^2x+\cos ^2x=1\)If we substitute the number 1 in our statement for this identity, we're going to have
\(\begin{gathered} 1-\cos ^2x=\sin ^2x \\ (\sin ^2x+\cos ^2x)-\cos ^2x=\sin ^2x \\ \sin ^2x+\cos ^2x-\cos ^2x=\sin ^2x \\ \sin ^2x=\sin ^2x \\ 1=1 \end{gathered}\)Since we got a true equation in the end, the statement on this item is TRUE.
item c)
\(\frac{1}{\sin x}=\csc x\)This is a given identity(you can check it is a part of the table above the question), then, it is TRUE.
item d)
\(\frac{\cos x}{\sin x}=\cos x(\frac{1}{\sin x})\)To solve this one, we're going to use the following property.
\(\frac{a}{b}=a\cdot\frac{1}{b}\)You can take out the numerator as a coefficient for our fraction.
Then, this statement is also TRUE.
item e)
\(\frac{\cos x}{\sin x}=\cos x\csc x\)Using the previous two statements, we can solve this one,
Since the statement d is true
\(\frac{\cos x}{\sin x}=\cos x(\frac{1}{\sin x})\)And statement c is also true
\(\frac{1}{\sin x}=\csc x\)Then, if we substitute c on d, we have statement e, therefore, statement e is also TRUE.
item f)
\(\cos ^2x-1=\sin ^2x\)
To solve this one, we're going to use the following identity
\(\cos ^2x+\sin ^2x=1\)Again, making a substitution, we have
\(\begin{gathered} \cos ^2x-1=\sin ^2x \\ \cos ^2x-(\sin ^2x+\cos ^2x)=\sin ^2x \\ \cos ^2x-\sin ^2x-\cos ^2x=\sin ^2x \\ -\sin ^2x=\sin ^2x \end{gathered}\)Since the last equation is false, then the statement f is also FALSE.
item g)
\(\frac{2}{\cos x}=2\sec x\)Using the same property used on item d, we have
\(\frac{2}{\cos x}=2(\frac{1}{\cos x})\)And also using the definition os sec x, we have
\(\sec x=\frac{1}{\cos x}\)Using those properties in our statement, we have
\(\begin{gathered} \frac{2}{\cos x}=2\sec x \\ 2(\frac{1}{\cos x})=2\sec x \\ 2(\sec x)=2\sec x \\ 2\sec x=2\sec x \end{gathered}\)Then, our last statement g is also TRUE.