Answer:
tan O = √3/ 3.
Step-by-step explanation:
The tangent is positive in quadrant 3.
The adjacent side has length √((2^2 - (-1)^2)
= -√3.
So tan O = -1 / -√3
= 1 / √3
= √3/ 3.
To determine tan(θ) for an angle θ where sin(θ) = -1/2 and θ is in quadrant III, we perform the following steps:
1. **Understand the given information:** When sin(θ) = -1/2, it means the opposite side to angle θ in a right triangle is -1 and the hypotenuse is 2. Since θ lies in the third quadrant, both sine and cosine values are negative, implying that the adjacent side length must also be negative.
2. **Use the Pythagorean theorem:** For a right triangle:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
Substituting hypotenuse and opposite side values:
\[ 2^2 = (-1)^2 + \text{adjacent}^2 \]
Simplify and solve for the adjacent side:
\[ 4 = 1 + \text{adjacent}^2 \]
\[ 3 = \text{adjacent}^2 \]
\[ \text{adjacent} = \sqrt{3} \] or \[ \text{adjacent} = -\sqrt{3} \]
Since the angle is in the third quadrant where cosine is also negative, we must take the negative square root:
\[ \text{adjacent} = -\sqrt{3} \]
3. **Calculate tan(θ):** By definition,
\[ \tan(θ) = \frac{\text{opposite}}{\text{adjacent}} \]
Substitute the known side lengths:
\[ \tan(θ) = \frac{-1}{-\sqrt{3}} \]
4. **Simplify the expression:**
\[ \tan(θ) = \frac{1}{\sqrt{3}} \]
To rationalize the denominator:
\[ \tan(θ) = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \]
\[ \tan(θ) = \frac{\sqrt{3}}{3} \]
Thus, if sin(θ) = -1/2 and θ is in quadrant III, then tan(θ) = √3/3. Note that in the context of trigonometry, it is often assumed that the angle and the ratios are dimensionless, so a negative 'opposite' simply reflects the direction along the coordinate axis rather than a literal negative length.
If f(x)=x^2+2x−5 and g(x)=2x+4, what is (f⋅g)(x)
Answer:
Step-by-step explanation:
f(x)=x²+2x−5 and g(x)=2x+4
(f⋅g)(x) = (x²+2x−5)(2x+4) = 2x^3 +4x² -10x +4x²+8x-20
(f⋅g)(x) = 2x^3+8x² -2x -20
For this case we have the following functions:
[tex]f (x) = x ^ 2 + 2x-5\\g (x) = 2x + 4[/tex]
We must find [tex](f * g) (x)[/tex]. By definition we have to:
[tex](f * g) (x) = f (x) * g (x)[/tex]
So:
[tex](f * g) (x) = (x ^ 2 + 2x-5) (2x + 4)[/tex]
Applying distributive property we have:
[tex](f * g) (x) = 2x ^ 3 + 4x ^ 2 + 4x ^ 2 + 8x-10x-20[/tex]
We add similar terms:
[tex](f * g) (x) = 2x ^ 3 + 8x ^ 2-2x-20[/tex]
Answer:
[tex](f * g) (x) = 2x ^ 3 + 8x ^ 2-2x-20[/tex]
Simplify (x-4)(x^2+5x+2).
Answer: x^3 + 1x^2 - 18x - 8
Step-by-step explanation: Using the box method, we get the values x^3, 5x^2, 4x^2, 20x, 2x, and 8. Then you add the like terms to get your final answer.
lol my diagram got glitched out when uploading, sorry if its hard to understand
Which of the given expressions results in 0 when evaluated at x = 5?
OA. 5x(x – 7)
OB. (x - 3)(x - 5)
Oc. (x + 7)(x - 2)
OD. (x + 5)(x - 8)
Answer:
B
Step-by-step explanation:
Look for an expression with (x - 5 ) as this will give zero when x = 5, that is
x - 5 = 5 - 5 = 0
The only expression with a factor of (x - 5 ) is B
Ronalds furniture house is currently making 50 sofas in a day. How many sofas will be produced if prudction is to be increased by 150%
Answer:
125
Step-by-step explanation:
50 x 150% = 75
75 + 50 = 125
Answer:
125
Step-by-step explanation:
50 × 150% = 50 × 1.5 = 75
75 + 50 = 125
125 sofas is the answer
In each of the following pairs, determine the temperature change, and whether the temperature increased or decreased. a. 45° F to –10° F b. 35° C to 80° F c. –19° C to 49° C d. 137° C to 0° C e. 5° C to 19° F f. 68° F to 29° C
A: decreased
B: increased
C: increased
D: decreased
E: decreased
F: increased
Answer:
∵ Change = final value - initial value,
If change > 0, then the value is increased,
if change < 0, then the value is decreased,
a. 45° F to -10° F,
Change = -10° F - 45° F = -55° F < 0° F,
Thus, temperature is decreased,
b. 35° C to 80° F
∵ 35° C = 95° F ( by the formula (°C × 9/5) + 32 = °F )
Change = 80° F - 95° F = -5° F < 0° F,
Thus, temperature is decreased,
c. -19° C to 49° C,
Change = 49° C - (-19)° C = 68° C > 0° C,
Thus, temperature is increased,
d. 137° C to 0° C,
Change = 0° C - 137° C = -137° C < 0° C,
Thus, temperature is decreased,
e. 5° C to 19° F,
∵ 5°C = 41°F
Change = 19° F - 41° F = -22° F < 0° F,
Thus, temperature is decreased,
f. 68° F to 29° C,
∵ 29°C = 84.2°F
Change = 84.2°F - 68° F = 16.2° F > 0° F,
Thus, temperature is increased.
estimate 16.667 + 7.66 by first rounding each number to the nearest tenth
Answer:
24.4
Step-by-step explanation:
16.667 + 7.66 becomes 16.7 + 7.7, which, in turn, is 24.4
Answer:
24.4
Step-by-step explanation:
The nearest tenth is the first decimal place. We look at the second decimal place. If it is 5 or above we round up
16.667 to the nearest tenth = 16.7
7.66 to the nearest tenth = 7.7
16.7 + 7.7 = 24.4
Triangle ABC has coordinates A(2, 4), B(5, 6), and C(1, 4). If the triangle is rotated 180° clockwise about the origin, what are the coordinates of B'?
Answer:
B'(-5,-6)
Step-by-step explanation:
The mapping for 180° clockwise rotation about the origin is:
[tex](x,y)\to (-x,-y)[/tex]
The find the image of B(5,6), we plug in x=5 and y=6 into the rule.
[tex](5,6)\to (-5,-6)[/tex]
Therefore the coordinates of B' are (-5,-6).
Point B with coordinates (5, 6) will have coordinates (-5, -6) after a 180° clockwise rotation about the origin.
Triangle ABC has coordinates A(2, 4), B(5, 6), and C(1, 4). If the triangle is rotated 180° clockwise about the origin, we need to find the coordinates of B'.
To rotate a point (x, y) 180° about the origin, we can multiply the coordinates by -1. So, the coordinates of B' after rotation will be (-5, -6).
F = {(-1, 2), (3, 2), (4, 2), (0, 2)} Is F a function and why/why not?
Yes, no two ordered pairs in this list have the same first coordinate.
No, each set of ordered pairs in this list has the same second coordinate.
Yes, there is more than one ordered pair in this list.
No, there is a limited number of ordered pairs in this list.
Answer:
Yes, no two ordered pairs in this list have the same first coordinate
Step-by-step explanation:
we know that
A function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.
That means that there can be only one ordered pair with the same value of x.
therefore
In this problem F is a function, because all the ordered pairs of the list don't have the same first coordinate
Please help ASAP
this table shows how many students from high schools attended a foot ball game.
A student is randomly selected.
what is the probability that a student attended the game, given that the student is from North Beach? Round your answer to two decimal places.
In total, there are 200 students from North Beach.
Of those 200 students, 110 attended the game.
Because it is only asking for the probability that a student from North Beach attended, the answer would be 110 / 200, or 0.55.
Therefore, the answer would be C.
Hope this helps! :)
Answer:
0.314
Step-by-step explanation:
Given:
total number of students=350
total number of students that attented= 170
students that attended from North beach =110
probability that a student attended the game, given that the student is from North Beach= 110/350
=0.314 !
What is the first term in this expansion?
1.) 2x^15
2.) 2^15x^15
3.) 10y^15
4.) 10^15y^15
Answer:
2¹⁵x¹⁵ (2nd option)
Step-by-step explanation:
When you expand (2x+10y)¹⁵ you would get:
32,768x¹⁵ + 2,457,600x¹⁴y + 86,016,000x¹³y² + 1,863,680,000x¹²y³+
27,955,200,000x¹¹y⁴+307,507,200,000x¹⁰y⁵ + 307,507,200,000x¹⁰y⁵ + ... so on and so forth.
Since you are not interested in anything else but the first term, observe that the first term is:
32,768x¹⁵
That is equal to: 2¹⁵x¹⁵
What happened there is you need to distribute the exponent to the coefficient of the term as well, so you need to distribute it to 2 as well.
Y’all what do i write
Answer:
y = - 0.5x + 4
Step-by-step explanation:
Calculate the slope m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (0, 4) and (x₂, y₂ ) = (4, 2) ← 2 points on the line
m = [tex]\frac{2-4}{4-0}[/tex] = [tex]\frac{-2}{4}[/tex] = - 0.5
Note the line crosses the y- axis at (0, 4) ⇒ c = 4
y = - 0.5x + 4 ← equation of boundary line
Answer:
y = -0.5 x + 4.
Step-by-step explanation:
Slope-intercept form of the equation of a line on a Cartesian Plane:
[tex]y = mx + b[/tex].
[tex]m[/tex] is the slope of the line.
If two different points [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] on the line are given, the slope of the line will equal
[tex]\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}[/tex].
For this line, the two points are:
[tex](4,2)[/tex], and [tex](0, 4)[/tex].As a result,
[tex]\displaystyle m = \frac{2 - 4}{4 - 0} = - \frac{1}{2}[/tex].
[tex]b[/tex] is the [tex]y[/tex]-coordinate (the second coordinate) of the point at the intersection of the line and the [tex]y[/tex]-axis (the vertical axis.) The [tex]x[/tex]-coordinate (the first coordinate) of that point shall equals [tex]0[/tex]. For this line, the coordinates of the intersection is [tex](0, 4)[/tex]. As a result, [tex]b = 4[/tex].
The slope-intercept form of the equation of this line will thus be
[tex]\displaystyle y = -\frac{1}{2}x + 4[/tex].
Express in exponential form. log 3 a = n
Answer:
[tex]\log_3(a)=n[/tex] has exponential form [tex]3^n=a[/tex].
[tex]\log(3a)=n[/tex] has exponential form [tex]10^{n}=3a[/tex].
Please let me know if neither of my interpretations of your problem/question is correct. That is I can either assume you wrote [tex]\log_3(a)=n[/tex] or [tex]\log(3a)=n[/tex].
If something else was intended, please let me know. Thanks kindly.
Step-by-step explanation:
Let's assume: [tex]\log_3(a)=n[/tex].
The base is 3.
The exponent is n. (Just remember the logarithm is the exponent.)
So we have [tex]3^n=a[/tex]
In general these forms are equivalent:
[tex]log_b(a)=y \text{ is equivalent to } b^y=a[/tex].
[tex]\log(3a)=n[/tex] is the same as [tex]\log_{10}(3a)=n[/tex]
The base is 10 (if you don't see a base and log is written).
The exponent is n.
So we have [tex]10^n=3a[/tex].
To transform the logarithmic function log3 a = n into an exponential form, we apply the rule that if logb a = n, then b^n = a. Therefore, the exponential form is 3^n = a.
Explanation:The student's question is about transforming a logarithmic equation into an exponential form. This is a common request in mathematics, particularly in algebra and precalculus.
The given equation is log3 a = n. The exponential form of a logarithmic equation is based on this relationship: If logba = n, then the exponential form is bn = a. Applying this rule to the provided equation gives 3n = a as the exponential form.
This transformation is based on the definition of logarithms, which are the inverses of exponential functions. The logarithm of a number to a particular base is the exponent to which the base must be raised to get that number. So in this case, n is the power you need to raise 3 to in order to get a.
Learn more about Logarithms here:https://brainly.com/question/37287488
#SPJ3
Which fraction is not equivalent to 5/6?
A-15/18
B-20/30
C-10/12
D-30/36
Answer: B. 20/30
Step-by-step explanation: 20/30 simplified to 2/3. Multiply each number by 2 to get 4/6. 4/6 is not the same number as 5/6.
Answer:
B. 20/30
Step-by-step explanation:
Let's investigate
You have an appointment at 2 o'clock at the dentist. Then you need to catch a
train one hour after your appointment. The walk to the train station takes
20 minutes. What time will you have to leave the dentist to catch your train?
The graph shows the function f(x) = |x – h| + k. What is the value of k? k = –2.5 k = –1 k = 1 k = 2.5
Answer:
k= - 2.5
Step-by-step explanation:
If the graph is the attached picture then according to this:
The graph is in V- shape. So if the corners of the graph are not at (0,0) than there must be a constant.
k is the constant which means that it is the amount the function is vertically shifted. The graph is shifted 2.5 units down, therefore the value of k is -2.5.
Thus option 1 is correct....
find the absolute error of the measurement 10 meters. explain its meaning
Answer:
.5 m
Step-by-step explanation:
You can use maximum possible error to calculate absolute error
Maximum possible error of is half the unit of measure
This means it is ±0.5 meters
In this case, you find the place value the measurement 10 meters is rounded to .This measurement was measured to the nearest meter, thus the measuring unit is 1 meter.
Here you use the maximum possible error as the absolute error.
This means the measurement is 10 ± .5
whole numbers to make 380 in four different ways.
Answer:
There are several ways to make 380 with whole numbers. For example:
100 + 80 = 380220 + 160 = 380190 + 190 = 3801 + 379 = 380Use the zero product property to find the solutions to the equation 6x^2- 5x = 56.
Answer:
[tex]\large\boxed{x=-\dfrac{8}{3}\ or\ x=\dfrac{7}{2}}[/tex]
Step-by-step explanation:
[tex]6x^2-5x=56\qquad\text{subtract 56 from both sides}\\\\6x^2-5x-56=0\\\\6x^2+16x-21x-56=0\\\\2x(3x+8)-7(3x+8)=0\\\\(3x+8)(2x-7)=0\iff 3x+8=0\ \vee\ 2x-7=0\\\\3x+8=0\qquad\text{subtract 8 from both sides}\\3x=-8\qquad\text{divide both sides by 3}\\\boxed{x=-\dfrac{8}{3}}\\\\2x-7=0\qquad\text{add 7 to both sides}\\2x=7\qquad\text{divide both sides by 2}\\\boxed{x=\dfrac{7}{2}}[/tex]
Which of the following is a function?
A. {(1,1), (2,5), (-2,5), (1,-1)}
B. {(-1,1), (-2,1), (-1,3), (-4,1)}
C. {(-1,1), (-1,2), (-1,3), (-1,4)}
D. {(-1,1), (-2,2), (-3,3), (-4,4)}
Answer:
D
Step-by-step explanation:
Reminder: A function returns ONLY ONE y value for each input value.
We immediately discard possible answer A, since 2 points have input 1 with different outputs.
Same for B: discard this.
Same for C: discard this.
D is a function. All of the inputs are unique (no duplication).
Simplify 3/4(1/2×-12)+4/5
Answer: [tex]\frac{3}{8}x-\frac{41}{5}[/tex] or [tex]\frac{3}{8}x-8\ \frac{1}{5}[/tex]
Step-by-step explanation:
Given the following expression:
[tex]\frac{3}{4}(\frac{1}{2}x-12)+\frac{4}{5}[/tex]
You can follow these steps in order to simplify it:
1. You need to apply Distributive property:
[tex]=\frac{3}{8}x-9+\frac{4}{5}[/tex]
2. And now you must add the like terms. Then:
[tex]=\frac{3}{8}x-\frac{41}{5}[/tex]
Therefore you get the following answer:
[tex]\frac{3}{8}x-\frac{41}{5}[/tex] or [tex]\frac{3}{8}x-8\ \frac{1}{5}[/tex]
Clyde simplified ( j^-8)^-2
ANSWER
[tex]{j}^{16} [/tex]
EXPLANATION
The given expression is
[tex]( {j}^{ - 8} )^{ - 2} [/tex]
To simplify this expression, we use the law of exponents.
Recall that:
[tex] {(a}^{m} )^{n} = {a}^{mn} [/tex]
We need to multiply the exponents to get:
[tex]( {j}^{ - 8} )^{ - 2} = {j}^{ - 8 \times - 2} [/tex]
[tex]( {j}^{ - 8} )^{ - 2} = {j}^{16} [/tex]
Therefore after simplification, Clyde got:
[tex]{j}^{16} [/tex]
Answer:
Clyde should have applied the power of a power rule and multiplied the exponents instead of adding them. The simplified form should be j to the 16th power.
Step-by-step explanation:
The exponents should be multiplied.
The simplified form should be j^16
Can you guys please help me with this two questions
Answer:
1. C) 5.50x + 10
2. 54
Step-by-step explanation:
1. C) 5.50x + 10
You have a static fee of $10 which does not need to be attached to a variable as it will not change. You also have a fee of $5.50, which may be charged depending on the amount of classes. Since this is dependent on something, it includes a variable.
Therefore, your terms are:
5.5x
10
Simply add them together for your equation.
[tex]5.5x+10[/tex]
2. 54
To solve, simply plug in 8 to your equation which you created in part 1. This is possible because x stands for the amount of classes and, as the question tells us, 8 is the amount of classes.
[tex]5.5(8)+10\\44+10\\54[/tex]
Betsy can clean a/b of the house in an hour. How long will it take her to clean the whole house?
Answer:
[tex]b/a\ hours[/tex]
Step-by-step explanation:
we know that
Betsy can clean a/b of the house in an hour
The whole house (100%) correspond to the number 1
so
using proportion
Find out how long will it take her to clean the whole house
[tex]\frac{(a/b)}{1}=\frac{1}{x} \\ \\x=1/(a/b)\\ \\x=b/a\ hours[/tex]
6. The sum of 3 consecutive odd integers is 87. Find the equation used to solve this
problem. Find the three integers.
An +n + 2 +n +4-87:
n-27, n + 2 - 29, n + 4- 31
B
n
+n +1 +n + 2 - 87:
n=28, n + 1 = 29, n + 2 = 30
Cn +n +1+n + 2 = 87;
n = 29, n + 1 = 30. n + 2 = 31
Dn+n + 2 +n + 4-87;
n = 29,0 + 2 = 31, n + 4 = 33
Answer:
A
n=27, n+2=29, n+4=31
The correct option is A: n + n + 2 + n + 4 = 87; n -27, n + 2 - 29, n + 4 - 31.
Let's denote the three consecutive odd integers as n, n + 2, and n + 4. The sum of these integers is given as 87.
The correct equation to represent this situation is:
n + (n + 2) + (n + 4) = 87
Now let's solve for n:
3n + 6 = 87
Subtract 6 from both sides:
3n = 81
Divide both sides by 3:
n = 27
n + (n + 2) + (n + 4) = 87
27 + 29 + 31 = 87
The three consecutive odd integers are 27, 29, and 31.
The volume of a cube depends on the length of its sides. This can be written
in function notation as ws). What is the best interpretation of 3) = 27?
Answer:
Step-by-step explanation:
The given question is that the volume of a cube depends on the length of its sides.This can be written in function notation as v(s). What is the best interpretation of v(3)=27.
Solution:
According to the question the volume of a cube depends on the length of its sides. According to the statement we will apply the formula of volume of a cube.
V(s)=s³
In this question we have given s=3ft.
So, we will put the value of 's' in the formula.
V(s)=s³
V(3)=3³
Multiply 3 three times to get the answer.
V(3)=3*3*3
V(3)=27 ft³
This means that the cube has a volume of 27ft³ with the length of its sides 3ft....
Given a triangle with a = 12, A = 13°, and B = 12°, what is the length of c? Round to the nearest tenth.
Answer:
=5.1 units
Step-by-step explanation:
To use the sine rule to find the length of c we first need to find the angle at C.
C=180-(13+12)
C=180-25
C=155°
a/Sin A=c/Sin C
Calculating for the value of c using the sine rule:
12/Sin 13=c/Sin 155
c=12 sin 155
=5.07 units
Thus to 1 decimal places, the value of c is 5.1 units.
Selena is graphing the inequality x <-2
She begins by drawing a number line and placing an open circle at –2. What’s Selena’s next step?
Answer:
My man above in Correct
Step-by-step explanation:
If R(x) = 2x - 5 and g(x) = x2 - 4x - 8, find (f + g(%).
O A (f+9)(x) = x2 - 2x-3
O B. (f+g)(x) = x2 - 2x-13
O C. (*+ g)(x) = 3x2 - 4x- 13
O D. (f+ g)(x) = x2 + 2x-3
C
PREVIOUS
Answer:
OB. (f+g)(x) = x^2-2x-13
Step-by-step explanation:
Given
f(x) = 2x-5
and
g(x) = x^2-4x-8
We have to find (f+g)(x)
So,
(f+g)(x) = f(x) + g(x)
= 2x-5 + (x^2-4x-8)
= 2x-5+x^2-4x-8
= x^2+2x-4x-5-8
=x^2-2x-13
Therefore, Option B is correct ..
Select the correct answer.
If a line crosses the y-axis at (0, 1) and has a slope of 4/5, what is the equation of the line?
Answer:
5y - 4x = 5. This gives you your points with a slope of 4/5
Answer:
[tex]y=\frac{4}{5}x+1[/tex]
Step-by-step explanation:
We are given the slope, and the y-intercept. So we can use the slope-intercept equation. Which is as follows:
[tex]y=mx+b[/tex]
where [tex]m[/tex] is the slope and [tex]b[/tex] is the y-intercept (the value of [tex]y[/tex] when [tex]x=0[/tex]).
The information that we have is that the slope of the line is:
[tex]m=\frac{4}{5}[/tex]
and that the line crosses the y-axis at (0, 1).
From this point we can find the y-intercept [tex]b[/tex], because b is the value of [tex]y[/tex] when [tex]x=0[/tex].
And since the line passes through (0, 1) when [tex]x = 0[/tex], [tex]y =1[/tex]. Thus, the y-intercept is:
[tex]b=1[/tex]
substituting the values of the slope and the y-intercept into the equation:
[tex]y=mx+b\\y=\frac{4}{5}x+1[/tex]
the answer is: [tex]y=\frac{4}{5}x+1[/tex]
p(x) is a polynomial with integer coefficients and p(-3) = 0. Which statements must be true? Choose all that apply. (MORE THAN ONE ANSWER)
x + 3 is a factor of the polynomial.
x - 3 is a factor of the polynomial.
-3 is the constant term of the polynomial.
p(x) can have at most 3 linear factors.
Answer:
x+3 is a factor of P
Step-by-step explanation:
By the factor theorem:
P(a)=0 <-> x-a is a factor of P.
You have P(-3)=0.
This implies x-(-3) is a factor of P.
Note: x-(-3)=x+3.