Answer:
-6Step-by-step explanation:
[tex]\text{The average rate of change of function}\ f(x)\\\text{over the interval}\ a\leq x\leq b\ \text{is given by this expression:}\\\dfrac{f(b)-f(a)}{b-a}\\\\f(x)=2x^2-7x\\\\\text{Put the values of x = 2 and x = 6 to the equation of the function:}\\\\f(2)=2(2^2)-7(2)=2(4)-14=8-14=-6\\f(6)=2(6^2)-7(6)=2(36)-42=72-42=-30\\\\\dfrac{f(6)-f(2)}{6-2}=\dfrac{-30-(-6)}{4}=\dfrac{-24}{4}=-6[/tex]
Ali, Ben and Joe sell 73 raffle tickets between them. Ben sells twice as many tickets as Ali, and Joe sells 8 more tickets than Ben. How many tickets does each person sell?
Answer:
Ali-13
Ben-26
Joe-34
Step-by-step explanation:
Ali-x
Ben-2x
Joe-2x+8
x+2x+2x+8=73
3x+2x+8=73
5x+8=73
5x=73-8
5x=65
x=65/5
=13
Ali sells 13 tickets
Ben sells 13×2
=26 tickets
Joe sells (13×2)+8
=34 tickets
Ali sells 13 tickets. Ben sells 26 tickets. Joe sells 34ctickets
How many tickets does each person sell?Let
Ali = x
Ben = 2x
Joe = 2x+8
⇒ x+2x+2x+8=73
⇒ 3x+2x+8=73
⇒ 5x+8=73
⇒ 5x=73-8
⇒ x=65/5
⇒ 13
Ali sells 13 tickets.
Ben sells 13×2
= 26 tickets.
Joe sells (13×2)+8
= 34 tickets.
An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
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Find the area.
A. 19 ft²
B. 30 ft²
C. 31 ft²
D. 84 ft²
Answer: D. 84 ft^2
Step-by-step explanation: The area of a trapezoid is (a+b)/2 x h. A and B are the bases, and h is the height. Plug in the numbers.
(8+16)/2 x 7 = 84
Your answer would be 84 ft^2. D.
The graph of F(x), shown below, resembles the graph of G(x)=x^2, but it has been changed somewhat.which of the following could be the equation of F(x)?
Answer:
Choice A. [tex]F(x) = -(x - 4)^{2} -3[/tex].
Step-by-step explanation:
Both F(x) and G(x) are quadratic equations. The graphs of the two functions are known as parabolas. All four choices for the equation of F(x) are written in their vertex form. That is:
[tex]y = a(x - h)^{2} + k[/tex], [tex]a \ne 0[/tex]
where
The point [tex](h, k)[/tex] is the vertex of the parabola, andThe value of [tex]a[/tex] determines the width and the direction of the opening of the parabola. [tex]a >0[/tex] means that the parabola opens upward. [tex]a <0[/tex] means that the parabola opens downwards. The opening becomes narrower if the value of [tex]a[/tex] increases.For the parabola G(x),
the vertex is at the point [tex](4, -3)[/tex], andthe parabola opens downwards.In other words,
[tex]h = 4[/tex],[tex]k = -3[/tex], and[tex]a \le 0[/tex].Hence choice A. [tex]F(x) = -(x - 4)^{2} -3[/tex].
The only equation that correctly represents the graph of F(x) as a vertically stretched and reflected version of the graph of G(x), shifted to the right by 4 units.
Option A. [tex]F(x)=-(x-4)^{2}-3[/tex] is correct.
To answer this question, we can consider the following transformations of the graph of [tex]G(x)=x^2$:[/tex]
Vertical stretch: [tex]$F(x)=kx^2$[/tex] for some positive constant k.
This will stretch the graph of G(x) vertically by a factor of k.
Vertical shift: [tex]$F(x)=x^2+h$[/tex] for some constant $h$.
This will shift the graph of G(x) up or down by $h$ units.
* **Horizontal shift: [tex]F(x)=(x+h)^2[/tex] for some constant $h$. This will shift the graph of G(x) to the left or right by h units.
* **Reflection: [tex]F(x)=-x^2[/tex]. This will reflect the graph of G(x) across the x-axis.
The graph of F(x) in the image appears to be a vertically stretched and reflected version of the graph of G(x), shifted to the right by 4 units. This suggests that the equation of F(x) could be of the form:
F(x)=-k(x-h)^2 for some positive constant k and some constant h.
To find the values of k and h, we can use the fact that the graph of F(x) passes through the points (-2,3) and (4,-3).
Substituting x=-2 and y=3 into the equation above, we get:
[tex]3=-k(-2-h)^2[/tex]
and substituting x=4 and y=-3 into the equation above, we get:
[tex]-3=-k(4-h)^2[/tex]
Dividing the second equation by the first equation, we get:
[tex]\frac{1}{3}=\frac{(-2-h)^2}{(4-h)^2}[/tex]
Taking the square root of both sides, we get:
[tex]\pm \frac{1}{\sqrt{3}}=\frac{-2-h}{4-h}[/tex]
Solving for $h$, we get:
[tex]h=4\pm \frac{4}{\sqrt{3}}[/tex]
Substituting this value of h into either of the original equations, we can solve for k.
For example, substituting [tex]$h=4+\frac{4}{\sqrt{3}}$[/tex] into the first equation, we get:
[tex]3=-k\left(-2-4+\frac{4}{\sqrt{3}}\right)^2[/tex]
Solving for $k$, we get:
[tex]k=3\cdot \left(-6+\frac{4}{\sqrt{3}}\right)^2[/tex]
Therefore, one possible equation for $F(x)$ is:
[tex]F(x)=-3\left(x-4+\frac{4}{\sqrt{3}}\right)^2[/tex]
Another possible equation for $F(x)$ is:
[tex]F(x)=-3\left(x-4-\frac{4}{\sqrt{3}}\right)^2[/tex]
These two equations are equivalent, since they both represent the same reflected and vertically stretched version of the graph of G(x), shifted to the right by 4 units.
Therefore, the answer to the question is:
[tex]A. F(x)=-(x-4)^{2}-3[/tex]
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What is the surface area of the right prism below?
Answer:
C. [tex]432\text{ units}^2[/tex]
Step-by-step explanation:
We have been given an image of a right prism. We are asked to find the surface area of our given prism.
The surface area of our given prism will be sum of area of two base right triangles and area of three rectangles.
[tex]SA=2B+Ph[/tex], where,
SA = Surface area,
B = Area of each base,
P = Perimeter of base,
h = Height of prism.
Let us find area of right base of prism.
[tex]B=\frac{1}{2}\times 6\times 8[/tex]
[tex]B=3\times 8[/tex]
[tex]B=24[/tex]
Perimeter of base will be [tex]6+8+10=24[/tex].
Upon substituting these values in surface area formula, we will get:
[tex]SA=2*24+24*16[/tex]
[tex]SA=24(2+16)[/tex]
[tex]SA=24(18)[/tex]
[tex]SA=432[/tex]
Therefore, the surface area of our given prism is 432 square units and option C is the correct choice.
If sales tax is 6% and you buy a shirt for $25 how much do you owe?
Answer:
26.50
Step-by-step explanation:
First we need to calculate the sales tax
25 * 6%
Changing to decimal from
25 * .06
1.50
Then we add the tax to the cost of the shirt
25+1.50
26.50
We owe 26.50
Which expression is equivalent to (a^-8b/a^-5b^3)^-3
Answer:
The correct option is A.
Step-by-step explanation:
The given expression is:
(a^-8b/a^-5b^3)^-3
We have to keep one thing in our mind that in a division, same base, the exponents are subtracted.
We will change the division into multiplication: The denominator will become numerator and the signs of the exponent become opposite.
(a^-8 *a^+5 * b *b^-3)^-3
=(a^-8+5 * b^1-3) ^-3
=(a^-3*b^-2)^-3
=(a)^-3*-3 (b)^-2*-3
= a^9 b^6
Thus the correct option is A....
Answer:
a^9b^6
Step-by-step explanation:
Find the average rate of change for the given function from x = 1 to x = 2.
1) -2
2) -1/2
3) 1/2
4) 2
Answer:
2
Step-by-step explanation:
The average rate of change from x=1 to x=2 is the same as finding the slope of a line at x=1 and x=2.
So we are going to need to corresponding y coordinates.
What y corresponds to x=1? y=3
What y corresponds to x=2? y=5
So we have the ordered pairs (1,3) and (2,5).
Line the points up vertically and subtract vertically then put 2nd difference over 1st difference.
(2 , 5)
-(1 , 3)
-----------
1 2
The average rate of change is 2/1 or just 2.
Now since we were asked to find the average rate of change given the function was a line, it really didn't matter what two points you used on that line.
Answer: Fourth option
[tex]m=2[/tex]
Step-by-step explanation:
If we call m the average change rate of a function between [tex]x_1[/tex] and [tex]x_2[/tex], then, by definition:
[tex]m=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]
In this case the function is the line shown in the graph. Then we look for the values of [tex]y = f (x)[/tex] for [tex]x = 1[/tex] and [tex]x = 2[/tex]
When [tex]x=1[/tex] then [tex]f(x)=3[/tex]
When [tex]x=2[/tex] then [tex]f(x)=5[/tex]
Therefore
[tex]m=\frac{5-3}{2-1}[/tex]
[tex]m=\frac{2}{1}[/tex]
[tex]m=2[/tex]
if m + 1 is an an even integer which one of the following must be an odd integer ?
A : m - 1
B : 2m - 2
C : 2m + 1
D : 2m + 2
To find which expression must be an odd integer when m + 1 is even, we analyze each option using substitution. The only option that results in an odd integer is 2m + 1.
If m + 1 is an even integer, then m must be an odd integer. Let's analyze each option to see which one of them must be an odd integer:
(A) m - 1: Since m is odd, subtracting 1 from an odd number will result in an even number. Therefore, (A) m - 1 is not an odd integer.(B) 2m - 2: If we substitute an odd integer for m, we get (2 * odd) - 2, which simplifies to an even number. Thus, (B) 2m - 2 is not an odd integer.(C) 2m + 1: Substituting an odd integer for m gives (2 * odd) + 1, which simplifies to an odd number. Therefore, (C) 2m + 1 is an odd integer.(D) 2m + 2: If we substitute an odd integer for m, we have (2 * odd) + 2, which simplifies to an even number. Hence, (D) 2m + 2 is not an odd integer.Therefore, the answer is (C) 2m + 1 as it is the only option that must be an odd integer.
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Final answer:
Option C, which is 2m + 1, must be an odd integer if m + 1 is an even integer because adding 1 to any even integer results in an odd integer.
Explanation:
Given that m + 1 is an even integer, we can determine which of the other expressions would therefore be an odd integer. We know that any even integer can be written in the form 2k, where k is an integer, because even integers are multiples of 2. Similarly, odd integers can be written as 2k + 1, where k is also an integer.
Given the options:
A: m - 1, if m + 1 is even, then m is odd, and subtracting 1 from an odd number results in an even number, so this option is not correct.B: 2m - 2, this is clearly even because it is a multiple of 2 regardless of the value of m, so this option is not correct.C: 2m + 1, if m is odd (since m + 1 is even), this will result in an odd number, so this option is correct.D: 2m + 2, like option B, this is also clearly even, so this option is not correct.Therefore, the answer is C: 2m + 1.
A graph of quadratic function y = f(x) is shown below.
What is the solution set of the quadratic inequality f(x) ≥ 0?
Answer: it’s B 0
Step-by-step explanation:
Write an equation: The sides of a rectangle are x and ( x - 4 ). The perimeter is 87
The perimeter of a rectangle, is all of the sides added up.
In a rectangle, there are two widths (right and left), and two lengths (up and down)
So:
Perimeter = 2 × width + 2 × length.
To get the equation we just substitute in the values that we are given:
length = (x)
width = (x - 4)
perimeter = 87
So just put these values into: perimeter = 2 × width + 2 × length:
87 = 2(x) + 2(x - 4)
_____________________________
Answer:
The equation is: 87 = 2(x) + 2(x - 4)
(note: this can be simplified down like so:
87 = 2x + 2x - 8
87 = 4x - 8
95 = 4x
23.75 = x
)
diameter of a circular garden pool is six feet. what is the area of the pool in algebra
For this case we have that by definition, the area of a circle is given by:
[tex]A = \pi * r ^ 2[/tex]
Where:
r: It is the radius of the circle
As data we have that[tex]r = 6 \ ft[/tex]
Then, replacing we have:
[tex]A = \pi * 6 ^ 2\\A = 36 \pi[/tex]
Taking [tex]\pi = 3.14[/tex]
A = 113.04
So, the pool area is [tex]113.04 \ ft ^ 2[/tex]
Answer:
[tex]113.04 \ ft ^ 2[/tex]
Answer:
9π ft²
Step-by-step explanation:
Use a calculator to evaluate 9 and the square root of three to the nearest ten thousandth. a. 27.1234 c. 44.9569 b. 6.7050 d. 5.1962 Please select the best answer from the choices provided
Answer:
Option D 5.1962
Step-by-step explanation:
we have
The number [tex]9[/tex] and the number [tex]\sqrt{3}[/tex]
I assume for the choices provided that the operation is calculate [tex]9[/tex] divided by [tex]\sqrt{3}[/tex]
so
[tex]\frac{9}{\sqrt{3}} =5.1962[/tex] (rounded to the nearest ten thousandth)
Answer:
C. 44.9569
Step-by-step explanation:
what's the answer to this?
(x+1)²=25
[tex](x+1)^2=25\\x+1=5 \vee x+1=-5\\x=4 \vee x=-6[/tex]
Find the value of x
Answer:
x = 40
Step-by-step explanation:
The three angles in a triangle add to 180 degrees
x + 30 + (3x-10) = 180
Combine like terms
4x +20 = 180
Subtract 20 from each side
4x+20-20 = 180-20
4x= 160
Divide each side by 4
4x/4 =160/4
x = 40
F(t)=-2t+5 f()=13
Please help
Answer:
-4
Step-by-step explanation:
So we are given f(t)=-2t+5.
We are asked to find the value a such that f(a)=13.
If f(t)=-2t+5, then f(a)=-2a+5. I just replaced old input,x, with new input, a.
So we want to solve f(a)=13 for a.
f(a)=13
Replace f(a) with -2a+5:
-2a+5=13
Subtract 5 on both sides:
-2a =8
Divide both sides by -2:
a =-4
Check:
Is -2t+5=13 for t=-4?
-2(-4)+5=8+5=13, so yep.
The value of t is -4 for which the function will give 13 as output.
What is a function?A mathematical relationship from a set of inputs to a set of outputs is called a function.
How to find the value of t so that the function gives 13 as output?The given function is F(t) = -2t+5
To get the value of t we should equate the function with 13.So, -2t + 5 = 13
⇒ -2t = 8
⇒ t = -4
So, for the value of t = -4 the given function will give the output 13
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Which system of equations has only one solution?
Answer:
The answer is the last option, D.
Step-by-step explanation:
A function with one solution can be added and have a variable and a constant leftover. The variables in the other three options cancel each other out, so none of them have one solution. The first and third equations have no solutions, and the second has infinitely many solutions. The fourth option has only one variable cancel out and still has a constant on the other side of the equation. The fourth option is correct.
Is f(x)=([tex]\frac{1}{e}[/tex])^x a exponential function?
Answer:
Yes.
Step-by-step explanation:
It can be noticed that 1/e can be written as e^(-1). Whenever a number or a variable is shifted from the numerator to the denominator or vice versa, the power of that number or that variable becomes negative. This means that 1/e = e^(-1). Since f(x) = (1/e)^x, f(x) can be written as:
f(x) = (e^(-1))^x.
Since there are two powers and the base is same, so both the powers will be multiplied. Therefore:
f(x) = e^(-x).
It can be seen that e is involved in the function and it has a negative power of x, so this means that f(x) is a decreasing exponential function!!!
If sin theta =7/25, use the Pythagorean identity to find cos theta
Answer:
[tex]\cos(\theta)=\pm \frac{24}{25}[/tex]
Step-by-step explanation:
I don't know where [tex]\theta[/tex] is so there is going to be two possibilities for cosine value, one being positive while the other is negative.
A Pythagorean Identity is [tex]\cos^2(\theta)+\sin^2(\theta)=1[/tex].
We are given [tex]\sin(\theta)=\frac{7}{25}[/tex].
So we are going to input [tex]\frac{7}{25}[/tex] for the [tex]sin(\theta)[/tex]:
[tex]\cos^2(\theta)+(\frac{7}{25})^2=1[/tex]
[tex]\cos^2(\theta)+\frac{49}{625}=1[/tex]
Subtract 49/625 on both sides:
[tex]\cos^2(\theta)=1-\frac{49}{625}[/tex]
Find a common denominator:
[tex]\cos^2(\theta)=\frac{625-49}{625}[/tex]
[tex]\cos^2(\theta)=\frac{576}{625}[/tex]
Square root both sides:
[tex]\cos(\theta)=\pm \sqrt{\frac{576}{625}}[/tex]
[tex]\cos(\theta)=\pm \frac{\sqrt{576}}{\sqrt{625}}[/tex]
[tex]\cos(\theta)=\pm \frac{24}{25}[/tex]
A right cylinder has a radius of 5 m and a height of 3 m. What is the volume of the cylinder?
Answer:
V = 75 pi m^3
or approximately
V =235.5 m^3
Step-by-step explanation:
The volume of a cylinder is given by
V = pi r^2 h
where r is the radius and h is the height
V = pi * 5^2 *3
V = 75 pi m^3
If we use 3.14 as an approximation for pi
V =235.5 m^3
Answer:
volume = 75(pi) m^3 (exactly)
volume = 235.6 m^3 (approximately)
Step-by-step explanation:
volume of cylinder = pi * radius^2 * height
volume = pi * (5 m)^2 * 3 m
volume = 75(pi) m^3 (exactly)
volume = 235.6 m^3 (approximately)
Given the expression, , choose the correct horizontal asymptote. y=x^2-8x+3/x^2+3x+1 None y = 0 y = 1 y = -2
Answer:
y = 1
Step-by-step explanation:
Given function is:
[tex]\frac{x^2+8x+3}{x^2+3x+1}[/tex]
In order to find the horizontal asymptote, the coefficients of highest degree variable of numerator and denominator are divided.
In our case,
both the numerator and denominator have 1 as he co-efficient of x^2
So the horizontal asymptote is y = 1/1
Hence, third option y=1 s correct ..
Final answer:
The correct horizontal asymptote for the given rational expression is y = 1, found by comparing the degrees and leading coefficients of the numerator and denominator.
Explanation:
For the expression y = (x² - 8x + 3) / (x² + 3x + 1), the correct horizontal asymptote is determined by looking at the degrees of the polynomials in the numerator and the denominator. When the degrees are the same, the horizontal asymptote is found by dividing the leading coefficients. In this case, both polynomials are of degree 2, and the leading coefficients are both 1. Thus, the horizontal asymptote is y = 1.
Solve for variable: 5z/8 – 1/3 = 2/5
Answer: z= .6
Step-by-step explanation:
Numerator: When you do 5•.6, you get 3, and then 3-1 equals 2.
Denominator: 8 - 3 = 5.
Therefore you get 2/5 which makes z = .6
PLEASE URGENT
Carrie is finding the set of even numbers within the set of prime numbers.
Of the sets described, which is the universal set?
all real numbers
all integers
prime numbers
even numbers
Answer:
Prime Numbers
Step-by-step explanation:
Two sets are being considered in the given statement:
Set of even numbersSet of prime numbersA universal set is defined as such a set from which all the other subsets are taken for a given case. Or in other words, a universal set is such a set to which all the subsets belong and this universal set contains all the elements being studied/considered.
For the given case, set of Even Numbers is the subset which is being taken from the set of prime numbers. The set of prime numbers contains all the elements which we are considering for this very scenario.
Therefore, the set of prime numbers is the universal set.
Answer:
c
Step-by-step explanation:
The Reed High School basketball program earned $7,115.03 in profits over the current season. Based on game attendance, it is determined that the girls' basketball team will receive 7% of the profits, while the boys' basketball team will receive 11% of the profits. Which is the best estimate of how much more the boys' team will receive in profits than the girls' team?
A.
$770
B.
$490
C.
$280
D.
$1,260
Answer:
c. $280
Step-by-step explanation:
first you times the total profits by the percent that the girls get then the percent the boys get to find out how much of the profit the girls and boys basketball teams each get, then you subtract the profits the boys get by the profits the girls get to find out how much more the boys get than the girls.
7,115.03 x .07 = 498.0521
7,115.03 x .11 = 782.6533
782.6533 - 498.0521 = 284.6012
the boys receive 284.6012 more in profits than the girls
so c. $280 would be the best estimate of how much more the boys team will receive
Write the Recursive rule for the geometric sequence
An=1072-1
8, 4, 2, 1, 1/2,...
[tex]\bf 8~~,~~\stackrel{8\cdot \frac{1}{2}}{4}~~,~~\stackrel{4\cdot \frac{1}{2}}{2}~~,~~\stackrel{2\cdot \frac{1}{2}}{1}~~,~~\stackrel{1\cdot \frac{1}{2}}{\cfrac{1}{2}} \\\\\\ a_n=\cfrac{1}{2}\cdot a_{n-1}\qquad \begin{cases} a_1=\textit{previous term}\\ a_n=\textit{current term}\\ a_1=\textit{first term}\\ \qquad 8 \end{cases}[/tex]
The graph below shows the function f(x)=x-3/x^2-2x-3 which statement is true
Answer:
The correct option is A.
Step-by-step explanation:
Domain:
The expression in the denominator is x^2-2x-3
x² - 2x-3 ≠0
-3 = +1 -4
(x²-2x+1)-4 ≠0
(x²-2x+1)=(x-1)²
(x-1)² - (2)² ≠0
∴a²-b² =(a-b)(a+b)
(x-1-2)(x-1+2) ≠0
(x-3)(x+1) ≠0
x≠3 for all x≠ -1
So there is a hole at x=3 and an asymptote at x= -1, so Option B is wrong
Asymptote:
x-3/x^2-2x-3
We know that denominator is equal to (x-3)(x+1)
x-3/(x-3)(x+1)
x-3 will be cancelled out by x-3
1/x+1
We have asymptote at x=-1 and hole at x=3, therefore the correct option is A....
Answer:
Its A. I just passed the final exam with 5 minutes left of the SESSION.
Step-by-step explanation:
A solid right pyramid has a hexagon all base with an area of 7.4 units the pyramid has a height of 6 units what is the volume of the pyramid
Answer:
14.8 units³
Step-by-step explanation:
The volume of a "pointed" solid, such as a cone or pyramid, is one-third the product of its base area and height:
V = (1/3)Bh
Filling in the given numbers and doing the arithmetic, we find the volume to be ...
V = (1/3)(7.4 units²)(6 units) = 14.8 units³
The volume of the pyramid is 14.8 cubic units.
Ryan is looking for the sum of the cube of a number, n, and 16.
What is the sum Ryan is looking for if n = 2?
Answer:
24
Step-by-step explanation:
Changing the word problem into an equation you get that you are trying to find n^3+16. This is because they are looking for the cube of a number n plus 16. Plugging in n as 2 you get 2^3+16 which is 8+16 which is 24.
Answer:
24
Step-by-step explanation:
PLEASE HELP!!!
Eliminate the parameter. x = 5 cos t, y = 5 sin t. Answer must be a rectangular equation
Answer:
[tex]x^2+y^2=25[/tex]
Step-by-step explanation:
Recall the following Pythagorean Identity:
[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]
Let's solve the x equation for cos(t) and the y equation for sin(t).
After the solve we will plug into our above identity.
x=5cos(t)
Divide both sides by 5:
(x/5)=cos(t)
y=5sin(t)
Divide both sides by 5:
(y/5)=sin(t)
Now we are ready to plug into the identity:
[tex]\sin^2(t)+\cos^2(t)=1[/tex]
[tex](\frac{y}{5})^2+(\frac{x}{5})^2=1[/tex]
[tex]\frac{x^2}{5^2}+\frac{y^2}{5^2}=1[/tex]
Multiply both sides by 5^2:
[tex]x^2+y^2=5^2[/tex]
This is a circle with center (0,0) and radius 5.
All I did to get that was compare our rectangular equation we found to
[tex](x-h)^2+(y-k)^2=r^2[/tex]
where (h,k) is the center and r is the radius of a circle.
Angle θ is in standard position. If (8, -15) is on the terminal ray of angle θ, find the values of the trigonometric functions.
Answer:
* sin Ф = -15/17 * cos Ф = 8/17 * tan Ф = -15/8
* csc Ф = -17/15 * sec Ф = 17/8 * cot Ф = -8/15
Step-by-step explanation:
Lets revise the trigonometric function of angle Ф
- Angle θ is in standard position
- Point (8, -15) is on the terminal ray of angle θ
- That means the terminal is the hypotenuse of a right triangle x and y
are its legs
∵ x-coordinate is positive and y-coordinate is negative
∴ angle Ф lies in the 4th quadrant
- The opposite side of angle Ф is the y-coordinate of the point on the
terminal ray of angle Ф and the adjacent side to angle Ф is the
x-coordinate of that point
∵ The length of the hypotenuse (h) = √(x² + y²)
∴ h = √[(8)² + (-15)²] = √[64 + 225] = √[289] = 17
∴ The length of the hypotenuse is 17
- Lets find sin Ф
∵ sin Ф = opposite/hypotenuse
∵ The opposite is y = -15
∵ The hypotenuse = 17
∴ sin Ф = -15/17
- Lets find cos Ф
∵ cos Ф = adjacent/hypotenuse
∵ The adjacent is x = 8
∵ The hypotenuse = 17
∴ cos Ф = 8/17
- Lets find tan Ф
∵ tan Ф = opposite/adjacent
∵ The opposite is y = -15
∵ The adjacent = 8
∴ tan Ф = -15/8
- Remember csc Ф is the reciprocal of sin Ф
∵ csc Ф = 1/sin Ф
∵ sin Ф = -15/17
∴ csc Ф = -17/15
- Remember sec Ф is the reciprocal of cos Ф
∵ sec Ф = 1/ cos Ф
∵ cos Ф = 8/17
∴ sec Ф = 17/8
- Remember cot Ф is the reciprocal of tan Ф
∵ cot Ф = 1/tan Ф
∵ tan Ф = -15/8
∴ cot Ф = -8/15
Amir stands on a balcony and throws a ball to his dog, who is at ground level.
The ball's height (in meters above the ground), xxx seconds after Amir threw it, is modeled by
h(x)=-(x+1)(x-7)
How many seconds after being thrown will the ball reach its maximum height?
________ seconds
I know I literally just posted one a couple minutes ago but, please help me again T-T
Answer:
3 seconds
Step-by-step explanation:
The given equation describes a parabolic curve with zeros at x=-1 and x=7. The line of symmetry of the curve is halfway between the zeros, at ...
x = (-1 +7)/2 = 6/2 = 3
The maximum of the curve is on the line of symmetry. The ball will reach its maximum height 3 seconds after being thrown.
_____
Additional comment
The maximum height will be 16 meters.
The ball will reach its maximum height 3 seconds after being thrown.
Explanation:To find the time when the ball reaches its maximum height, we need to determine the vertex of the quadratic equation representing the ball's height. The equation h(x)=-(x+1)(x-7) can be rewritten in vertex form as h(x)=-x^2+6x-7. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -1 and b = 6. Plugging these values into the formula, we get x = -6/-2 = 3. Therefore, the ball reaches its maximum height 3 seconds after being thrown.