The graph of the parent quadratic
function f(x) = x2 and that of a second function
of the form g(x) = ax2 are shown. What
conclusion can you make about the value of a in the equation of the second function?
Answers
The correct answer is D. The value of a in the equation of the second function is greater than 1.
Describe Quadratic Function?A quadratic function is a type of function in algebra that can be written in the form f(x) = ax² + bx + c, where a, b, and c are constants (coefficients) and x is the independent variable. The graph of a quadratic function is a parabola, which is a U-shaped curve.
The coefficient a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The point where the parabola changes direction is called the vertex, which is located at the point (-b/2a, f(-b/2a)).
The coefficient b determines the horizontal position of the vertex, and c determines the vertical position of the vertex.
We know that the parent quadratic function f(x) = x² is a U-shaped curve with its vertex at the origin (0,0).
If the graph of a second function g(x) = ax² intersects the graph of f(x) at exactly one point, then this point must be the vertex of g(x), since the vertex of f(x) is already fixed at the origin.
In order for g(x) to intersect f(x) at exactly one point, the vertex of g(x) must lie on the x-axis (since it cannot be above the x-axis, or else there would be two points of intersection). Therefore, the y-coordinate of the vertex of g(x) is 0.
The vertex of a quadratic function of the form ax² + bx + c has x-coordinate -b/2a and y-coordinate c - b²/4a. Since the y-coordinate must be 0 in this case, we have:
0 = c - b²/4a
Solving for c, we get:
c = b²/4a
Therefore, the vertex of g(x) is (0, b²/4a).
We can see from the graph that the vertex of g(x) is to the right of the origin, so its x-coordinate is positive. This means that the coefficient a must be positive, since otherwise the parabola would open downwards and the vertex would be below the x-axis.
We also know that the vertex of g(x) is above the x-axis, so its y-coordinate (which is b²/4a) must be positive. This means that b^2 and a have the same sign.
Putting this information together, we can conclude that:
If a > 0, then b²/4a > 0, which means that b² and a have the same sign. This means that the parabola opens upwards, and the vertex is above the x-axis. From the graph, we can see that this is the case, so we can eliminate options B and C.
If a < 1, then the parabola is narrower than the parent function f(x) = x², and the vertex is closer to the y-axis. This is not the case from the graph, so we can eliminate option A.
Therefore, the correct answer is D. The value of a in the equation of the second function is greater than 1.
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The complete question is:
Related Questions
For what values of a are the following expressions true?
1/2a - 5 1/2 = 5-a
Answers
Therefore, the values of a that make the expression true are approximately 0.378 and 14.121.
What is expression?In mathematics, an expression is a combination of numbers, variables, and operations that represents a quantity or a mathematical relationship. Expressions can be simple or complex, and can involve arithmetic, algebraic, trigonometric, or other mathematical operations. Expressions can also include constants, which are fixed values that do not change. Expressions can be evaluated by substituting numerical values for the variables and performing the indicated operations.
Here,
To solve for the values of a that make the given expressions true, we need to isolate the variable a on one side of the equation. We can start by simplifying the left side of the equation:
1/2a - 5 1/2 = 5 - a
Multiplying both sides by 2a gives:
1 - 10a* 1/2 = 10a - 2a²
Simplifying the left side by multiplying both terms by 2:
2 - 20a = 10a - 2a²
Rearranging the terms gives a quadratic equation:
2a² - 30a + 2 = 0
We can solve for a by using the quadratic formula:
a = (-b ± √(b² - 4ac)) / 2a
where a = 2, b = -30, and c = 2.
Plugging in these values gives:
a = (30 ± √((-30)² - 4(2)(2))) / 4
a = (30 ± √(892)) / 4
a ≈ 0.378 or a ≈ 14.121
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The square ABCD is reflected over
the y-axis then translated five units
down and one unit to the left to
make the square A'B'C'D'.
Question: If the original coordinate
pair for the point A is (-5, 3), what is
the coordinate pair of A'?
Answers
The coordinate pair for A' is as a result (4, -2) in coordinated square.
After reflecting the square ABCD over the y-axis and translating it five units down and one unit to the left, we may use the following transformations in order to get the coordinate pair of A':
Across the y-axis, reflect the square: This modifies the x-sign coordinate's while maintaining the y-original coordinate's value. Hence, A's new coordinates would be (5, 3).
Five units down and one unit to the left, translate the square as follows: The y-coordinate is moved five units down and the x-coordinate one unit to the right as a result. A's new system coordinates are calculated to be (4, -2).
the coordinate pair for A' is as a result (4, -2).
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Line A has equation y = -4x + 1 and line B contains the points (-2,1) and (1,-11). Select any statement(s) below that are true.
Line B has a slope of -5.
Line A has a slope of -4.
Line A and line B have the same slope.
Line B has a flatter slope than line A.
Line A has a flatter slope than line B.
Answers
Answer:
Line A has a slope of -4
Line A and line B have the same slope
Step-by-step explanation:
Line A is in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. As we can see, -4 is m in line A so -4 is the slope of line A.
If we have two points on a line, we can find the slope using the slope formula
[tex]m = \frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex], where x1 and y1 are one of the points and x2 and y2 come from the other point.
Thus, if we allow (-2, 1) to be x1 and y1 and (1, -11) to be x2 and y2,
[tex]m=\frac{-11-1}{1-(-2)}\\ m=-4[/tex]
Since the slope of Line B is -4, it has the same slope as line A
PROBLEM SOLVING WITH TREND LINES
2. 75
2. 50
Part I: The scatter plot at the left shows the
cost of gas per gallon during certain years.
Use the scatter plot to answer questions 1-5.
2. 25
2. 00
2017-1970 = ?
1. 75
1. 50
PRICE OF GAS PER GALLON ($)
1. 25
Y=0. 05(?) +0. 25
1. 00
. 75
. 50
. 25
5
10
15 20 25 30 35 40 45 50
YEARS (SINCE 1970)
Answers
The trend line equation y = 0.05x + 0.25 can be used to accurately predict the cost of gas per gallon for any year since 1970.
The equation of the trend line in the scatter plot is given by y = 0.05x + 0.25, where x is the number of years since 1970 and y is the price of gas per gallon in dollars. This equation can be used to calculate the price of gas in a given year since 1970. For example, if we want to calculate the price of gas in 2017, we can plug in x = 47 (the number of years since 1970) into the equation to get y = 0.05(47) + 0.25 = 2.85. This means that in 2017, the price of gas per gallon was approximately 2.85 dollars. This equation can be used to accurately predict the cost of gas for any given year since 1970.
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SOMEONE HELP PLEASE I'LL GIVE BRAINLIEST IF BOTH OF THESE ARE CORRECT
Simplify.
a)√(30+10√5)
b) √(25+10√5+5)
Answers
1. Therefore, √(30+10√5) simplifies to 5 + √5.
2. Therefore, √(25+10√5+5) simplifies to 5 + √5 as well.
Step by step solved:
a) To simplify √(30+10√5), we can try to identify a perfect square within the radicand. We see that 5 is a perfect square, so we can rewrite the expression as follows:
√(30+10√5) = √(5(6+2√5)) = √5 * √(6+2√5)
Now, we can try to find a number that, when squared, gives us 6+2√5. We can see that (√5 + 1)^2 = 6 + 2√5, so we can rewrite the expression as:
√5 * √(6+2√5) = √5 * (√5 + 1) = 5 + √5
Therefore, √(30+10√5) simplifies to 5 + √5.
b) To simplify √(25+10√5+5), we can simplify the radicand first by combining like terms:
√(25+10√5+5) = √(30+10√5)
Now, we can use the same process as in part a) to simplify the expression to 5 + √5.
Therefore, √(25+10√5+5) simplifies to 5 + √5 as well.
help please i don’t get it
Answers
[tex]\stackrel{ \textit{multiplying the 1st equation by 9} }{\begin{array}{lllllll} 9(~x&+y&=& ~~ 0)\\\\ -9x&-5y&=&-24 \end{array}}\hspace{5em} \begin{array}{llll} ~~ 9x&+9y&=&0\\\\ -9x&-5y&=&-24\\\cline{1-4} ~~0&-4y&=&-24 \end{array} \\\\[-0.35em] ~\dotfill\\\\ -4y=-24\implies y=\cfrac{-24}{-4}\implies \boxed{y=6} \\\\\\ \stackrel{\textit{substituting on the 1st equation}}{x+(6)=0\implies \boxed{x=-6}} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill (-6~~,~~6)~\hfill[/tex]
what rattional numbers are between -5.25 and -5.26
Answers
Answer:
-5.255
Step-by-step explanation:
Read the excerpt from "John Burns of Gettysburg” by Bret Harte. And it was terrible. On the right Raged for hours the heady fight, Thundered the battery's double brass,— Difficult music for men to face; While on the left—where now the graves Undulate like the living waves That all the day unceasing swept Up to the pits the rebels kept Which use of consonance best conveys the violent sounds of the battle? “And it was terrible. On the right” “Thundered the battery’s double brass” “Difficult music for men to face” “Undulate like the living waves”
NEED HELP ASAP! I WILL GIVE BRAINLIEST TO WHO ANSWERS FIRST
Answers
The use of consonance in "Thundered the battery's double brass" best conveys the violent sounds of the battle.
What is consonance?
Consonance is a literary device where consonant sounds are repeated in close proximity, often at the ends of words, to create a musical or rhythmic effect in a text.
The passage uses consonance, which is the repetition of consonant sounds within or at the end of words, to convey the violent sounds of the battle.
In particular, the repeated "t" and "d" sounds in "Thundered the battery's double brass" and "Difficult music for men to face" create a harsh, jarring effect that emphasizes the chaos and danger of the fighting.
Similarly, the repeated "v" and "l" sounds in "Undulate like the living waves" create a sense of movement and fluidity, conveying the unrelenting and overwhelming force of the rebel advance.
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Christine swam 2 kilometers against the current in the same amount of time it took her to swim 8 kilometers with the current. The rate of the current was 3 kilometers per hour. How fast would Christine swim if there were no current?
Answers
If the rate of the current was 3 kilometers per hour, Christine's speed in still water is 5 kilometers per hour.
Let's start by assuming that Christine's speed in still water is x kilometers per hour. We know that the current is flowing at a rate of 3 kilometers per hour.
When Christine is swimming against the current, her effective speed is reduced by the speed of the current, so her speed is (x-3) kilometers per hour. If it takes her the same amount of time to swim 2 kilometers against the current as it does to swim 8 kilometers with the current, we can set up an equation:
time to swim 2 km against current = time to swim 8 km with current
We can use the formula distance = speed x time to express the distance that Christine covers in each scenario. The distance she covers against the current is 2 kilometers, and the distance she covers with the current is 8 kilometers. So our equation becomes:
2 / (x - 3) = 8 / (x + 3)
To solve for x, we can cross-multiply and simplify:
2(x + 3) = 8(x - 3)
2x + 6 = 8x - 24
6x = 30
x = 5
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Jefferson high school has an enrollment of 1,864 students and 564 graduate how much will still be in school
Answers
Answer:
1300
Step-by-step explanation:
1864-564= 1300 students
(total - no. graduated)
Hope this helps! :)
Write an explicit formula for the sequence.
a^n+1=an+7, a^1=−22
A. a^n=7−22n
B. a^n=−22+7(n−1)
C. a^n=7−22(n−1)
D. a^n=−22+7n
Answers
Answer:
[tex]\textsf{B)} \quad a_n=-22+7(n-1)[/tex]
Step-by-step explanation:
An explicit formula for a sequence allows you to find the nth term of the sequence.
A recursive formula for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.
Given recursive rule:
[tex]\begin{cases}a_{n+1}=a_n+7\\a_1=-22\end{cases}[/tex]
From the given recursive rule, we can see that each term is 7 more than the previous term. Therefore, the common difference between terms is d = 7.
Explicit formula[tex]a_n = a_1 + (n - 1)d[/tex]
Input the values of a₁ and d into the explicit formula:
[tex]\implies a_n=-22+(n-1) \cdot 7[/tex]
[tex]\implies a_n=-22+7(n-1)[/tex]
Therefore, the explicit formula for the given sequence is:
[tex]\boxed{a_n=-22+7(n-1)}[/tex]
OFFERING BRAINLIEST TO STEP BY STEP EXPLANATION
Answers
Answer:
[tex]112cm^2[/tex]
Step-by-step explanation:
Volume of a rectangluar prism = Length x Width x Height
Volume, Length, and Width is given now we can solve for Height, c
[tex]V = l * w * h[/tex]
[tex]80m^3 = 4m * 4m * h[/tex]
[tex]h = 80m^3 / 16m^2[/tex]
[tex]h = 5m[/tex]
Surface area of a rectangular prism is A = 2(wl +hl +hw)
[tex]A = 2(4m * 4m + 5m * 4m + 5m * 4m)[/tex]
[tex]= 2(16m^2 + 20m^2 + 20m^2)[/tex]
[tex]= 2(56cm^2)\\A= 112cm^2[/tex]
Hope this helps!
Brainliest is much appreciated!
supongo q es es 8 cm pero no creo q sería 2 o l
The reading on the thermometer is -10 degrees celsius and the temperature drops with 10 degrees celsius what will the new reading be
Answers
The thermometer reads -10 degrees Celsius, and the temperature drops by 10 degrees Celsius. As a result, the new thermometer reading will be -20 degrees Celsius.
If the temperature drops by 10 degrees Celsius and the initial reading on the thermometer is -10 degrees Celsius, we can calculate the new reading as follows:
New reading = Initial reading - Temperature drop
New reading = (-10) - 10
New reading = -20
Therefore, the new reading on the thermometer will be -20 degrees Celsius.
When we say the temperature drops by 10 degrees Celsius, we mean that the temperature decreases by 10 degrees Celsius from its initial value. In this case, the initial value is -10 degrees Celsius.
So, if the temperature drops by 10 degrees Celsius, we need to subtract 10 from the initial value to get the new temperature reading. This is because a decrease in temperature means the temperature value becomes lower.
The resulting new reading of -20 degrees Celsius is 10 degrees Celsius lower than the initial reading of -10 degrees Celsius. It indicates that the temperature has decreased further, and it is now even colder than before.
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Please answer with full explanation!
Answers
Answer: 5cm
Step-by-step explanation:
You divide the area of green (93.75) by pink's length (7.5)
that gives you 12.5.
12.5 is the length of green.
12.5 - 7.5 = 5
You subtract the 7.5 because to get yellow's length because Yellow's length plus pink's length is equal to green's total length.
The density of some steel is 7.85 g/cm³. What is the mass of 50 cm³ of this steel? Give your answer to 1 d. p.
Answers
Answer:
392.5g
Step-by-step explanation:
Density =mass/volume
Mass = Density x volume
Mass = 7.85g/cm³ x 50 cm³
Mass = 392.5g
you are called to investigate a possible health concern in an office building that may be due to an environmental toxin. you arrange with a company that occupies two levels of the building to follow all 300 of their office employees. you intend to follow all employees for 30 days and every day, the employees report their health status. however, on day 10, 35 employees report respiratory distress. on day 20, 70 employees report respiratory distress. the remaining employees do not report any respiratory distress over the 30-day period.
Answers
The investigation of a possible health concern in an office building due to an environmental toxin involves following the health status of all employees for a period of time, identifying potential sources of toxins, and taking steps to remove the toxins and protect the health of the employees.
The possible health concern in an office building that may be due to an environmental toxin can be investigated by following all 300 of their office employees for 30 days and observing their health status. On day 10, 35 employees report respiratory distress and on day 20, 70 employees report respiratory distress.
The remaining employees do not report any respiratory distress over the 30-day period. This suggests that there may be an environmental toxin in the office building that is causing respiratory distress in the employees.
It is important to further investigate the source of the toxin and take steps to remove it in order to protect the health of the employees. This may include conducting air quality tests, examining the building for potential sources of toxins, and implementing measures to improve ventilation and air quality.
It is also important to continue monitoring the health of the employees and providing them with appropriate medical care if needed.
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Using trigonometry, work out the length y. Give your answer in centimetres to 1 d.p. y 6.1 cm 39° Not drawn accurately
Answers
The length of the opposite side (y) is equal to 3.2 centimeters.
How to calculate the length y?In order to determine the length of the opposite side (y), we would apply the sine trigonometry ratio because the given side lengths represent the opposite side and hypotenuse of a right-angled triangle.
cos(θ) = Adj/Hyp
Where:
Opp represent the opposite side of a right-angled triangle.Hyp represent the hypotenuse of a right-angled triangle.θ represent the angle.Based on the information provided in the image, we can logically deduce the following parameters:
Opp = y cm.Hyp = 6.1 cm.Angle, θ = 39°.By substituting the parameters into the sine trigonometry ratio formula, we have the following;
sin(θ) = Opp/Hyp
sin(39) = y/5
y = 5sin(39)
y = 3.2 centimeters.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
What is the equation of the line?
Answers
Answer:
Use the slope formula and slope-intercept form
y=mx+b to find the equation.
the steps and answers are in the photos
{I hope this helps you :)
the ratio of the sides of a triangle is 3:10:7. What is the length of the shorest side?
Answers
Answer:
The length of the shortest side is 3 units.
Step-by-step explanation:
Let's assume that the shortest side has a length of 3x, where x is a constant.
Then, the other sides would have lengths of 10x and 7x, respectively.
To check if this is a valid triangle, we need to make sure that the sum of the lengths of any two sides is greater than the length of the third side:
3x + 10x > 7x (sum of shortest and middle side is greater than longest side)
3x + 7x > 10x (sum of shortest and longest side is greater than middle side)
10x + 7x > 3x (sum of middle and longest side is greater than shortest side)
Simplifying these inequalities, we get:
13x > 7x
10x > 3x
17x > 0
All of these inequalities are true if x > 0, so our assumption is valid.
Therefore, the length of the shortest side is 3x. To find the value of x, we can set up the following equation based on the given ratio:
3x : 10x : 7x = 3 : 10 : 7
Simplifying this equation by dividing all sides by x, we get:
3 : 10 : 7 = 3 : 10 : 7
This is true, so any value of x will satisfy the given ratio.
However, to find the length of the shortest side, we can simply substitute x = 1 into our assumption:
Shortest side = 3x = 3(1) = 3
Therefore, the length of the shortest side is 3 units.
Hope this helps! Sorry if this is wrong. :]
______are set of repeated activitieswhich have well defined results
Answers
Answer:
statistical experiments performed on a well defined sample space
Step-by-step explanation:
experiments can be performed either by probability, or other repetitive actions on sample space for possible outcomes.
Show your work for every single one plssssss
Answers
The values would be
[tex]15. = \frac{1}{1000y^3}\\\\16. =-\dfrac{1024}{n^5}\\\\17. =\dfrac{1}{32k^{10}}\\\\18. =$\dfrac{49}{36c^2}$\\\\\\19. = $64^t$\\\\20. = $5 \cdot 5^t$\\\\21. = $81 \cdot 3^{4t}$\\\\22. = $\dfrac{2^{2t}}{2}$ or $2^{2t-1}$[/tex]
What is an expression?
In mathematics, an expression is a combination of symbols and/or numbers that represents a value or a relationship between values. Expressions can be as simple as a single number or variable, or they can be complex, involving multiple operations and functions. Expressions can be used to describe relationships between variables, to represent mathematical formulas or equations, and to perform calculations.
[tex]15. = \frac{1}{1000y^3}\\\\16. =-\dfrac{1024}{n^5}\\\\17. =\dfrac{1}{32k^{10}}\\\\18. =$\dfrac{49}{36c^2}$\\\\\\19. = $64^t$\\\\20. = $5 \cdot 5^t$\\\\21. = $81 \cdot 3^{4t}$\\\\22. = $\dfrac{2^{2t}}{2}$ or $2^{2t-1}$[/tex]
Hence, the values would be
[tex]15. = \frac{1}{1000y^3}\\\\16. =-\dfrac{1024}{n^5}\\\\17. =\dfrac{1}{32k^{10}}\\\\18. =$\dfrac{49}{36c^2}$\\\\\\19. = $64^t$\\\\20. = $5 \cdot 5^t$\\\\21. = $81 \cdot 3^{4t}$\\\\22. = $\dfrac{2^{2t}}{2}$ or $2^{2t-1}$[/tex]
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I need help on this asap!
Answers
If Mr Corazon produces the model that would give the most profit only, this idea would be incorrect. The reason is that that poor people would not be able to buy the product.
What is profit maximization?
Profit maximization is the process by which a business or a firm determines the optimal level of output or production that leads to the highest possible profit. It is an important goal for most firms, as it helps to ensure their long-term sustainability and competitiveness in the market.
Based on the question, at the profit of $375 a piece, it would mean that the price has become too high. Less people would but the product. This makes his idea a bad one.
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the radius of a sphere is increasing at a rate of 4 mm/s. how fast is the volume increasing (in mm3/s) when the diameter is 100 mm? (round your answer to two decimal places.)
Answers
The volume is increasing at a rate of 16755.21 mm³/s when the diameter is 100 mm.
We are given that the radius of a sphere is increasing at a rate of 4 mm/s.
Let's first write the formula of the volume of a sphere;
[tex]V=43πr3V =\frac{4}{3}\pi r^3V =34πr3[/tex]
Now we can differentiate it by using the chain rule;
[tex]V'=43π3r2drdt+4πr3(4)orV' =4πr2drdt[/tex]
The problem asks for how fast the volume is increasing, so we need to find V' when the diameter is 100 mm, or r=50 mm.
Thus;
[tex]V'=4π(50²)(4) =40,000πV' =125663.71 mm³/s[/tex]
Therefore, the volume is increasing at a rate of 125663.71 mm³/s when the diameter is 100 mm. Now we round to two decimal places to get;V'=16755.21 mm³/s.
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A certain baseball pitcher gives up a base hit (on average) once every 15 pitches. in a particular game he threw 120 pitches. find the probability he gave up at MOST 2 base hits.
Answers
The probability that the pitcher gave up at most 2 base hits in the game is approximately 0.53.
What is probability?
Probability is a branch of mathematics in which the chances of experiments occurring are calculated.
Let X be the number of base hits given up in 120 pitches. Then X follows a binomial distribution with parameters n = 120 (number of trials) and p = 1/15 (probability of success in each trial).
The probability of giving up at most 2 base hits can be calculated as follows:
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
Using the binomial probability formula, we have:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where (n choose k) is the binomial coefficient, which gives the number of ways to choose k successes from n trials.
Plugging in the values, we get:
P(X = 0) = (120 choose 0) * (1/15)^0 * (14/15)^120 ≈ 0.028
P(X = 1) = (120 choose 1) * (1/15)^1 * (14/15)^119 ≈ 0.17
P(X = 2) = (120 choose 2) * (1/15)^2 * (14/15)^118 ≈ 0.33
Therefore,
P(X ≤ 2) = 0.028 + 0.17 + 0.33 ≈ 0.53
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PLEASE HELP I NEED HELP ASAP
Answers
Answer:
Sin= 35/37
cos= 13/37
tan= 35/13
Step-by-step explanation:
Evaluate the expression 7x, if x = 11.
18
77
4
711
Answers
[tex] \tt \: c)77[/tex]
_________
[tex] \tt{7x} \\ \: \: \: \: \: = \tt 7(11) \\ \tt= 77[/tex]
[tex] \: [/tex]
━━━━━━━━━━━━━━━━━━━━━━━
hope it helps☘️
(1). A cyclist starts a journey from town A. He rides 10km north, then 5km east and finally 10km on a bearing of 045°. a) How far east is the cyclist's destination from town A? b). How far north is the cyclist's destination from town A? c). Find the distance and bearing of the cyclist's destination from town A (Correct your answers to the nearest km and degree)
Answers
The required answers are a) 17.1 km b) 7.1 km, c) 18.6 km away from town A on a bearing of 293°.
How to find the distance and angle?To solve this problem, we can use vector addition to find the displacement vector from town A to the cyclist's destination.
a) To find how far east the cyclist's destination is from town A, we need to find the east component of the displacement vector. We can break down the displacement vector into its north and east components using trigonometry:
[tex]$$\text{East displacement} = 5\text{ km} + 10\text{ km}\cos(45^\circ) = 5\text{ km} + 10\text{ km}\frac{\sqrt{2}}{2} = 10\text{ km} + 5\sqrt{2}\text{ km} \approx 17.1\text{ km}$$[/tex]
So the cyclist's destination is approximately 17.1 km east of town A.
b) Similarly, to find how far north the cyclist's destination is from town A, we need to find the north component of the displacement vector:
[tex]$$\text{North displacement} = 10\text{ km}\sin(45^\circ) = 10\text{ km}\frac{\sqrt{2}}{2} = 5\sqrt{2}\text{ km} \approx 7.1\text{ km}$$[/tex]
So the cyclist's destination is approximately 7.1 km north of town A.
c) To find the distance and bearing of the cyclist's destination from town A, we can use the Pythagorean theorem and trigonometry. The displacement vector is the hypotenuse of a right triangle with legs of length 17.1 km and 7.1 km, so its length is:
[tex]$$\text{Displacement} = \sqrt{(17.1\text{ km})^2 + (7.1\text{ km})^2} \approx 18.6\text{ km}$$[/tex]
To find the bearing of the displacement vector, we can use the inverse tangent function:
[tex]$$\text{Bearing} = \tan^{-1}\left(\frac{\text{East displacement}}{\text{North displacement}}\right) \approx 67^\circ$$[/tex]
However, this angle is measured clockwise from north, so we need to subtract it from 360° to get the bearing measured counterclockwise from north:
[tex]$$\text{Bearing} = 360^\circ - 67^\circ = 293^\circ$$[/tex]
So the cyclist's destination is approximately 18.6 km away from town A on a bearing of 293°.
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I bought a new car to transport my kids and all their stuff. The value of the car y (in thousands of dollars) can be represented by the model y=41(0. 92)^t
, where t is the number of years since I purchased the car. Which value in the equation represents the decay factor?
Answers
The decay factor in the equation y=41(0.92)^t is the value of (0.92) raised to the power of t.
In this case, the base of the exponential function is 0.92, which is less than 1. This means that as t increases, the value of (0.92)^t decreases exponentially, leading to a decay in the value of y. The decay factor is the factor by which the value of y decreases each year.
In this case, the decay factor is 0.92, which means that the value of the car decreases by 8% each year. This is because (0.92) x 100% = 92%, so the value of the car decreases to 92% of its previous value each year.
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Which of the japanese islands do you think became the center of power in japan
Answers
The Japanese island that became the center of power in Japan is Honshu.
Honshu is the largest and most populous island in Japan and has historically been the center of political, economic, and cultural power in Japan. It is home to Tokyo, the capital city of Japan, as well as many other major cities such as Osaka, Kyoto, and Yokohama. Honshu's central location and natural resources have made it a strategic location for political power throughout Japan's history, including during the feudal period when various powerful clans vied for control. Honshu's position as the economic and cultural hub of Japan has also helped solidify its status as the center of power, as it has been a center for education, trade, and innovation for centuries.
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can anyone explain this?
Two sides of a rectangle differ by 3.5cm find the dimensions of the rectangle if it's perimeter is 67cm.solve by matrix inversion method.
Answers
Using matrix inversion method and perimeter of rectangle, the dimension of the rectangle are 16cm and 7cm
What is the dimension of the rectangleLet the length of the rectangle be l and the width be w. We know that l - w = 3.5 and 2l + 2w = 67. We can rewrite these equations as a system of linear equations in matrix form:
[tex]$$\begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix} \begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}3.5 \ 67/2\end{bmatrix}$$[/tex]
We can solve for [tex]$\begin{bmatrix}l \ w\end{bmatrix}$[/tex] by multiplying both sides of the equation by the inverse of the coefficient matrix:
[tex]$$\begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix}^{-1} \begin{bmatrix}3.5 \ 67/2\end{bmatrix}$$[/tex]
To find the inverse of the coefficient matrix, we can use the following formula:
[tex]$$\begin{bmatrix}a & b \ c & d\end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix}d & -b \ -c & a\end{bmatrix}$$[/tex]
Plugging in the values for our matrix, we get:
[tex]$$\begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix}^{-1} = \frac{1}{1 \cdot 2 - (-1) \cdot 2} \begin{bmatrix}2 & 1 \ -2 & 1\end{bmatrix} = \begin{bmatrix}1/2 & 1/4 \ -1/2 & 1/4\end{bmatrix}$$[/tex]
Now we can substitute this matrix and the vector on the right-hand side of the equation into our formula to obtain the solution:
[tex]$$\begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}1/2 & 1/4 \ -1/2 & 1/4\end{bmatrix} \begin{bmatrix}3.5 \ 67/2\end{bmatrix} = \begin{bmatrix}16 \ 7\end{bmatrix}$$[/tex]
Therefore, the dimensions of the rectangle are 16 cm by 7 cm.
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