The notation "t(x)" represents the statement t is a function of x
How to interpret the notation?The notation is given as:
t(x) = ....
The above notation is a function notation, and it used to relate one element to another
For instance:
f(x) means f is a function of x
Similarly, "t(x)" represents the statement t is a function of x
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A 16 -foot ladder is leaning on a tree. The bottom of the ladder on the ground at a distance of 6 feet from the base of the tree. The base of the tree and the ground form a right angle. What is the distance, in feet, between the ground and the top of the ladder? Round your answer to the nearest tenth.
Answer:
14.8 feet
Step-by-step explanation:
since this is creating a right triangle you can use the Pythagorean theorem which is a^2 + b^2 = c^2, and by plugging in the one leg length and the hypotenuse length you get 14.8324 which rounds up to 14.8
The distance between the ground and the top of the ladder is 14.8 feet.
What is Pythagoras theorem?The relationship between the three sides of a right-angled triangle is explained by the Pythagoras theorem, commonly known as the Pythagorean theorem. The Pythagoras theorem states that the square of a triangle's hypotenuse is equal to the sum of its other two sides' squares.
Given:
Hypotenuse= 16 foot,
Base = 6 feet
Using Pythagoras theorem,
H² = B² + P²
16² = 6² + P²
256 = 36 +P²
P² = 256-36
P= 14.83 feet
Hence, the distance between the ground and the top of the ladder is 14.8 feet.
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In the diagram of AABC shown below, DE || BC.
B
A
3) 3
4) 15
E
If AB= 10, AD = 8, and AE = 12, what is the
length of EC?
1) 6
2) 2
C
dd or create
k as done
nments
As
to Susan Guha
Answer:
3
Step-by-step explanation:
AD/AB = AE/AC
8/10 = 12/AC
AC = 12(10/8)=15
EC = 15 - 12 = 3
Using the model from the example above, where f(x)=10001+999e−0.6030x, estimate the number of cases of flu on day 15. Remember that answers should be rounded to the nearest whole number.
The number of cases of flu on day 15 will be 1001 flus
Exponential functions are expressed as y = ab^x
Given the expression below
[tex]f(x)=10001+999e^{-0.6030x}[/tex]
In order to determine the number of cases of flu on day 15, we will substitute x = 15 into the expression to have:
[tex]f(x)=10001+999e^{-0.6030(15)}\\f(x)=10001+999e^{-9.045}\\f(15) = 10001+0.11786\\f(15) \approx 10001[/tex]
Hence the number of cases of flu on day 15 will be 1001 flus
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Answer:
Step-by-step explanation:
Evaluate the function for the value x=15.
f(x)f(15)f(15)=10001+999e−0.6030x=10001+999e−0.603(15)≈894.565217
Rounded to the nearest whole number, there will be 895 cases on day 15.
choose the equation below that represents the line passing through the point (-2,-3) with a slope of -6
The equation of the line is: y + 3 = -6(x + 2) [point-slope form] or y = -6x - 15 [slope-intercept form].
How to Write the Equation of a Line?Given a point (-2, -3) and a slope (m) of -6, you can write the equation of the line in point-slope form by substituting (a, b) = (-2, -3) and m = -6 into y - b = m(x - a):
y - (-3) = -6(x - (-2))
y + 3 = -6(x + 2) [point-slope form]
Rewrite in slope-intercept form as:
y + 3 = -6(x + 2)
y + 3 = -6x - 12
y = -6x - 12 - 3
y = -6x - 15 [slope-intercept form]
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The function below models the correlation between the number of hours a plant is kept in sunlight (x) and the height(y), in mm, to which it grows:
Y=2+4x
What does the y- intercept of this function represent?
the original height of the plant was 2mm. The correct option is the second one.
What does the y-intercept of the function represent?
We know that the function:
y = 2 + 4x
Represents the height of a plant in mm, as a function of x, the hours kepts in sunlight.
The y-intercept is the value that y takes when we evaluate in x = 0.
So, the y-intercept is the height of the plant when it is exposed to zero hours of light.
In this case:
y = 2 + 4*x
Evaluating in x = 0
y = 2 + 4*0 = 2
This means that the original height of the plant was 2mm. The correct option is the second one.
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A house has increased in value by 37% since it was purchased. If the original value was $720, 000, what is the value after the increase? A house has increased in value by 37 % since it was purchased . If the original value was $ 720 , 000 , what is the value after the increase ?
Answer:$986,4000
Step-by-step explanation:
Find f(5) for f(x)=1/4(2)^x
A. 2
B. 4
C. 32
D. 8
Step-by-step explanation:
The value of X is given so put the value of X in 1/4 (2)^x
You have learned how to divide polynomials in three different ways: by factoring, by long division, and by synthetic division. Explain in words and with mathematical representations how you would divide this polynomial in each of the three ways:
a) Factoring:
b) Long division:
c) Synthetic division:
d) Now, explain a connection you can see for each pair of methods. How does factoring relate to long division? How does long division relate to synthetic division? How does synthetic division relate to factoring?
The way to solve the polynomial using the methods is given below.
How to explain the polynomial?Polynomial by factoring:
An example is (t² + 5t + 4) / +t + 4)
= (t + 1)(t + 4)/(t + 4)
= t + 1.
Dividing polynomial by long division:
Divide the first term of the dividend by the first term of the divisor.
Put this as the first term in the quotient.
Subtract to create a new polynomial.
Continue until you can't divide anymore.
Dividing polynomial by synthesic division:
Step 1: Set up the synthetic division.
Step 2: Bring down the leading coefficient to the bottom row.
Step 3: Multiply c by the value just written on the bottom row.
Step 4: Add the column created in step 3.
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What is the value of h(a) when a = 2 in the
piecewise function
3a + 2
4a +5
3
h(a) =
when a > 3
when a = 3
when a <3
=========================================================
Explanation:
This is the given piecewise function
[tex]h(a) = \begin{cases}3a+2 \text{ when } a > 3\\4a+5 \text{ when } a = 3\\3 \text{ when } a < 3\\\end{cases}[/tex]
We can rewrite things like this
[tex]\text{when } a > 3, \ \ h(a) = 3a+2\\\text{when } a = 3, \ \ h(a) = 4a+5\\\text{when }a < 3, \ \ h(a) = 3\\[/tex]
I personally prefer this format better because it mentions the inputs first, and that leads to which function definition we go for.
The input a = 2 fits the description of [tex]a < 3[/tex] since [tex]2 < 3[/tex] is a true statement. Therefore we go for the third line and can see that [tex]h(a) = 3[/tex] becomes [tex]h(2) = 3[/tex]
-----------
As another example, if we had the input a = 7, then we'd go for the first line (since 7 > 3) and get
h(a) = 3a+2
h(7) = 3(7)+2
h(7) = 23
The information below explains the transformations and translations of a logarithmic function. What is the equation for this function?
Shift 4 units left
Shift 3 units up
Vertical Compress by 1/3
Horizontal stretch by 3
Fill in the table below for the information of the function f(x) = log[-(x – 5)] + 4
Function
Log[-(x - 5)] + 4
Domain
Range
Interval of Increase/Decrease
Equation of Asymptote
The transformed function is y = 1/3(log(x/3 + 4)) + 1
How to transform the logarithmic function?The parent logarithmic function is
y = log(x)
When shifted left by 4 units, we have:
y = log(x + 4)
When shifted up by 3 units, we have:
y = log(x + 4) + 3
When compressed vertically by 1/3, we have:
y = 1/3(log(x + 4) + 3)
This gives
y = 1/3(log(x + 4)) + 1
When stretched horizontally by 3, we have:
y = 1/3(log(x/3 + 4)) + 1
Hence, the transformed function is y = 1/3(log(x/3 + 4)) + 1
The transformation of function f(x)We have:
f(x) = log[-(x – 5)] + 4
Set the radicand greater than 0
-(x - 5) > 0
Divide by -1
x - 5 < 0
Add 5 to both sides
x < 5 -- this represents the domain
A logarithmic function can output any real number.
So, the range is [tex]-\infty < f(x) < \infty[/tex]
In the domain, we have:
x < 5
This means that the interval of decrease is [tex]-\infty < x < 5[/tex]
Rewrite as an equation
x = 5 --- this represents the equation of asymptote
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write an equation in slope- intercepty form of the line that passes through the point (5,9) with slope 7
Calculate the volume of a cylinder of radius 5cm and height 14cm (take pie 22/7)
Answer:
Volume of a cylinder = 1,100cm³
Step-by-step explanation:
Volume of a cylinder = πr²h
Given
π = 22/7
r = 5cm
h = 14cm
Substitute the given information in the problem.
Volume of a cylinder = 22/7 × (5cm)² × 14cm
Volume of a cylinder = 22/7 × 25cm² × 14cm
Volume of a cylinder = 1,100cm³
If f(x) = x..........
Answer:
b.) 0
Explanation:
⇒ f(g(-1))
go from right to left while solving such function questions.
⇒ f(-1 + 2)
⇒ f(1)
⇒ (1)² -2(1) + 1
⇒ 0
Step-by-step explanation:
[tex]f(x) = {x}^{2} - 2x + 1[/tex]
[tex]f(x) = {(x - 1)}^{2} [/tex]
[tex]g(x) = x + 2[/tex]
[tex] \: [/tex]
[tex] = f(g( - 1))[/tex]
[tex] = {(x - 1)}^{2} [/tex]
[tex] = {((x + 2) - 1)}^{2} [/tex]
[tex] = {(( - 1 + 2) - 1)}^{2} [/tex]
[tex] = {(1 - 1)}^{2} [/tex]
[tex] = {0}^{2} [/tex]
[tex] = 0[/tex]
The answer is B.
Which linear function represents the line given by the point-slope equation y-8=(x-4)?
A) f(x) = x +4
B) f(x) = x+6
C) f(x)=x-10
D) f(x)=x-12
The answer is (A) f(x) = x+4.
Use the function below to find F(4).
Answer:
A. 81
Step-by-step explanation:
F(x)=[tex]3^{x}[/tex]
if x=4, then F(4) = [tex]3^{4}[/tex] = 3×3×3×3=81
the cone shaped game piece is made of metal with a mass of 9.2g/cm3 what is the mass of the game piece
The volume of the piece of the metal Gina is found to be 22.82 cubic cm.
What is volume?Volume is defined as the space occupied by any object in the three-Dimensions. All three parameters are required for the volume like length, width and height of cube or cuboid.
Given the density of the metal = 9.2g/cm3
Given the mass of the piece of the metal = 210 g
Assuming the volume of a piece of metal to be V
The expression for the density is shown below:-
Density = Mass / Volume
9.2 = 210 / V
V = 210 / 9
V = 22.82 cubic cm.
Therefore the volume of the piece of the metal Gina is found to be 22.82 cubic cm.
The complete question is given below:-
Gina was given an irregularly shaped piece of metal and was told that the density of the metal was 10 g/cm3. She measured the mass of the piece to be 210 g. What was the volume of the piece of metal Gina was given?
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Given that sin(90-x)°=0.8. where x is an acute angle.Find without using calculator or mathematical table the value of tan x
By using trigonometric properties, we will see that:
tan(x) = 3/4.
How to find the value of tan(x)?
Remember that the sine of the difference is:
sin(a - b) = sin(a)*cos(b) - sin(b)*cos(a).
Here we have:
sin(90° -x ) = 0.8
Expanding the sine:
sin(90° - x) = sin(90°)*cos(x) - sin(x)*cos(90°)
We know that:
sin(90°) = 1 and cos(90°) = 0, then:
sin(90° - x) = sin(90°)*cos(x) - sin(x)*cos(90°) = cos(x).
Then we have:
cos(x) = 0.8
And we want to find the tangent of x, which is:
[tex]tan(x) = \frac{sin(x)}{cos(x)}[/tex]
We already know that cos(x) = 0.8
Now, remember that:
[tex]sin(x)^2 + cos^2(x) = 1\\\\sin(x) = \sqrt{1 - cos^2(x)} \\[/tex]
Because x is an acute angle, sin(x) > 0, that is why we choose the positive root.
Replacing cos(x) we get:
[tex]sin(x) = \sqrt{1 - 0.8^2} = \sqrt{1 - 0.64} = \sqrt{0.36} = 0.6[/tex]
Then the tangent of x is:
[tex]tan(x) = \frac{0.6}{0.8} = \frac{6}{10}*\frac{10}{8} = 6/8 = 3/4[/tex]
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Evaluate x² + 4x + 1 when x = -3
Answer: -2
Step-by-step explanation:
We can combine the fundamentals of mental math and the brain to calculate the problem when x is -2. You wake up noticing x is -2 and you instantly know the answer and solved in less than a second.
The table below shows which hand is favored by each person from a sample of 100 people (50 male and 50 female). Use the information in the table to answer the following questions
Determine the probability that a randomly selected person from the sample will be female or left-handed.
The probability that a randomly selected person from the sample will be female or left-handed will be 0.05.
What is the probability?Probability is synonymous with possibility. It is concerned with the occurrence of a random event.
Probability can only have a value between 0 and 1. Its simple notion is that something is very likely to occur. It is the proportion of favorable events to the total number of events.
Given data;
Total number of sample,n(S) = 100 person
No of female or left-handed is,n(E) =5
P(E) is the probability that a randomly selected person from the sample will be female or left-handed
The probability that a randomly selected person from the sample will be female or left-handed is found as;
P(E) = n(E)/n(S)
P(E) = 5 / 100
P(E) = 0.05
Hence the probability that a randomly selected person from the sample will be female or left-handed will be 0.05.
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A table has an area of x² + 12x + 35. Find the possible dimensions of the table.
Answer:
x + 7 and x + 5 are possible dimensions
Step-by-step explanation:
x² + 12x + 35 ← factor the quadratic
consider the product of the factors of the constant term (+ 35) which sum to give the coefficient of the x- term (+ 12)
the factors are + 7 and + 5 , since
7 × 5 = + 35 and 7 + 5 = + 12 , then
x² + 12x + 35 = (x + 7)(x + 5) ← in factored form
now the area A = length × breadth , that is
A = x² + 12x + 35 = (x + 7)(x + 5) , then
length = x + 7 or x + 5
breadth = x + 5 or x + 7
Which expression is equivalent to -1/6y+1/3?
An expression is defined as a set of numbers, variables, and mathematical operations. The correct option is C.
What is an Expression?In mathematics, an expression is defined as a set of numbers, variables, and mathematical operations formed according to rules dependent on the context.
The expression that is equivalent to -(1/6)y + 1/3 is,
-(1/6)y + 1/3
Take 1/6 common from both first and the last term,
1/6 (-y + 2)
Hence, the correct option is C.
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Solving Quadratic Inequalities
Nathan is a product manager for a home security company. Through testing and focus groups, he has modeled the equation P = -(x - 32)2 + 1000 to represent the profit (in thousands) of the price x for a new home security camera. Find the prices in dollars at which Nathan can make a profit of at least $471, 000. Show your work as it is done in the lesson content and include units of measurement.. HINT: Use 471 as part of the inequality you set up, not 471,000.
The prices in dollars at which Nathan can make a profit of at least $471, 000 are $55 and $9
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given the profit (P) in thousands for security camera with price of x, hence for a profit of at least $471, 000:
-(x - 32)² + 1000 ≥ 471
-(x² - 64x + 1024) + 1000 ≥ 471
-x² + 64x - 1024 + 1000 - 471 ≥ 0
-x² + 64x - 495 ≥ 0
x = 55 or x = 9
The prices in dollars at which Nathan can make a profit of at least $471, 000 are $55 and $9
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A total of 316 tickets were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was three times the number of adult tickets sold. How many adult tickets were sold?
Step-by-step explanation:
ATTACHED IS THE SOLUTION!Answer:
Adults: 79 tickets
Students: 237 tickets
Step-by-step explanation:
a + b = 316 Eq. 1
b = 3a Eq. 2
a = adults
b = students
Replacing Eq. 2 in Eq. 1
a + 3a= 316
4a = 316
a = 316/4
a = 79
from the Eq. 2
b = 3a
b = 3*79
b = 237
Check:
from the Eq. 1
a + b = 316
79 + 237 = 316
Use the function below to find F(3).
Answer: B
Step-by-step explanation:
Plug in x for 3. Then solve.
4*(1/3)^3
4*1/9
4/9
Which expression is equivalent to (3x² + 4x-7)(x-2)?
OA. (3x2 + 4x-7)(x) + (3x² + 4x-7)(-2)
OB. 2x(3x² + 4x-7)
OC. x(3x² + 4x-7) - 2
OD. x(3x² + 4x-7)+2(3x² + 4x-7)
Answer: A
Step-by-step explanation:
The expressions are equivalent by the distributive property.
Find the coordinates of the intersection of the diagonals of the parallelogram with the vertices W(−2, 5), X(2, 5), Y(4, 0),
and Z(0, 0)
.
Using the midpoint formula, the coordinates of the intersection of the diagonals of the parallelogram is: (1, 2.5).
What are Diagonals of a Parallelogram?The diagonals of a parallelogram bisect each other, therefore, the coordinates of their intersection can be determined using the midpoint formula, which is: [tex]M(\frac{x_2 + x_1}{2}, \frac{y_2 + y_1}{2})[/tex].
A diagonal is XZ.
X(2, 5) = (x1, y1)
Z(0, 0) = (x2, y2)
Plug in the values
[tex]M(\frac{0 + 2}{2}, \frac{0 + 5}{2})[/tex]
= (1, 2.5).
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write this ratio as a fraction in simplest form without any units. 3 pounds to 54 ounces
The ratio of the measurements written as a fraction is 8/9.
What are the ratios written as fraction?A fraction is a quantity that is not a whole number. In maths, a fraction usually has a numerator and a denominator.
In order to convert the ratio to fraction, the units of measurements have to be the same. So, pounds would be converted to ounces.
3 x 16 = 48 ounces
48 / 54 = 8/9
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If you rolled two dice, what is the probability
Give
that you would roll a sum of 5?
your answer as a simplified fraction. Enter
the
number that belongs in the green box.
second
first
1 2
3 456
1 1,1 1,2 1,3 1,4 1,5 1,6
2 2,1 2,2 2,3 2,4 2,5 2,6
3 3,1 3,2 3,3 3,4 3,5 3,6
§
4 4,1 4,2 4,3 4,4 4,5 4,6
5 5,1 5,2 5,3 5,4 5,5 5,6
6 6,1 6,2 6,3
6,4 6,5 6,6
Answer:
Probability of rolling a sum of 5 is 1/9.
There are 36 different combinations because 6(first roll)*6(second roll) = 36. The only possible combinations are {1,4},{2,3}{2,3}and{4,1}
4/36 = 1/9
10 + (-8) - 2
Answer the equation
Answer:
0
Step-by-step explanation:
[tex]10 + ( - 8) - 2 \\ = 1 0 - 8 - 2 \\ = 2 - 2 \\ = 0[/tex]
Answer: 0
Step-by-step explanation: -8+-2 is -10 and -10+10 is 0
Which of the following segments is a radius of OK?
M
K
OA. KM
B. MR
C. RT
D. MT
SUB
Answer:
A. KM
Step-by-step explanation:
A circle is named by its center. Circle K has its center at point K.
A radius is a line segment that extends from the center of the circle to a point on the circle. Point K must be one end of any radius.
ApplicationSegment KM is a radius.
Segments between points on the circle are chords. The other named segments in the answer choices are all chords of the circle.