Answer:
f^(-1)(x) = ±√(x - 5).
Step-by-step explanation:
Replace f(x) with y: y = x^2 + 5.
Swap the x and y variables: x = y^2 + 5.
Solve the equation for y. To do this, we'll rearrange the equation:
x - 5 = y^2.
Take the square root of both sides (considering both positive and negative square roots):
±√(x - 5) = y.
Swap y and x again to express the inverse function:
f^(-1)(x) = ±√(x - 5).
Given the piecewise functions shown below, select all of the statements that are true.
The true statements are:
a. f(-1) = 2
c. f(1) = 0
Let's evaluate each statement using the given piecewise function f(x):
a. f(-1) = -(-1) + 1 = 2
b. f(-2) = -(-2) + 1 = 3 (Not 0, so this statement is false)
c. f(1) = (1)^2 - 1 = 0
d. f(4) = (4)^2 - 1 = 16 - 1 = 15 (Not 7, so this statement is false)
Therefore, the correct statements are:
a. f(-1) = 2
c. f(1) = 0
Statement a is true because when x = -1, we use the first piece of the piecewise function, which gives us -(-1) + 1 = 2.
Statement c is true because when x = 1, we use the third piece of the piecewise function, which gives us (1)^2 - 1 = 0.
Statements b and d are false because they do not match the corresponding values obtained from evaluating the piecewise function at the given inputs.
Therefore, the true statements are:
a. f(-1) = 2
c. f(1) = 0
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6/7 .r = 3/4 write it as a fraction or as a whole or a mixed number
The solution for "r" is 21/24, which can be simplified to 7/8. The answer can be written as a fraction of 7/8.
To solve for "r" in the given equation, we can use algebraic manipulation.
First, we can multiply both sides of the equation by the reciprocal of 6/7, which is:
7/6: 6/7 · r = 3/4
7/6 · 6/7 · r = 7/6 · 3/4
r = 21/24.
Thus, the solution for "r" is 21/24, which can be simplified to 7/8.
Therefore, the answer can be written as a fraction of 7/8.
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Does the mapping diagram represent a function? Why or why not?
-5
8
9
y
-8
A. Yes; each input pairs with only one output.
B. No; the input value x = -5 pairs with two output values.
C. No; each input pairs with only one output.
D. No; each output value pairs with two input values.
The correct answer is: A. Yes; each input pairs with only one output.
To determine whether the given mapping diagram represents a function, we need to analyze the relationship between the input values (x) and the corresponding output values (y).
Looking at the mapping diagram, we can see that the input value -5 is paired with the output value 8. This implies that when x = -5, the corresponding y value is 8. Similarly, when x = 9, the corresponding y value is -8.
Since each input value has a unique and specific output value, we can conclude that the mapping diagram represents a function. In other words, for every input value, there is only one output value associated with it.
This is consistent with the definition of a function, where each input has a single and distinct output. Therefore, the mapping diagram satisfies the criteria for a function, and the correct answer is:
A. Yes; each input pairs with only one output.
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The mapping diagram does not represent a function that reflects this is: No; each output value pairs with two input values. D.
To determine whether the mapping diagram represents a function, we need to assess whether each input value pairs with only one output value.
Looking at the given mapping diagram:
-5 -> 8
9 -> y
-8 -> y
We can see that the input value -5 maps to the output value 8.
There are two different input values, 9 and -8, that map to the same output value, y.
In a function, each input should correspond to exactly one output, but in this case, the output value y is associated with two different input values.
In this case, y, the output value, pairs with both 9 and -8, indicating that the diagram does not meet the criteria for a function.
We must examine if each input value couples with only one output value in order to evaluate whether the mapping diagram reflects a function.
Observing the provided mapping diagram:
-5 -> 8 9 -> y -8 -> y
As can be seen, the input value of -5 corresponds to the output value of 8.
The identical output value, y, is mapped to two distinct input values, 9 and -8.
A function should have only one output for each input, however in this situation, the output value y is connected to two separate input values.
This figure does not fit the definition of a function because the output value, y, couples with both 9 and -8.
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two points A and B, due to two spheres X and Y 4.0m apart, that are carrying charges of 72mC and -72mC respectively. Assume constant of proportionality as 9×10^9Nm²/C². Find the electric field strength at points A and B due to each spheres presence
Due to the presence of spheres X and Y, the electric field strength at point B is [tex]1.01 * 10^6 N/C[/tex] and [tex]-4.05 * 10^6 N/C[/tex], respectively.
Given that two spheres X and Y are carrying charges of 72mC and -72mC respectively, and they are located 4.0 m apart from each other. The electric field strength at points A and B due to the presence of each sphere is to be determined.
Let's begin by calculating the electric field strength at point A due to sphere X. Electric field strength is given by E=kq/r², where k is Coulomb's constant, q is the charge and r is the distance between the two charges. The electric field strength at point A due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.)
Similarly, the electric field strength at point A due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (4.0m)^2 = 4.05 * 10^6 N/C[/tex] (approx.). Here, the negative sign indicates that the electric field due to sphere Y is in the opposite direction to the electric field due to sphere X. Now, let's calculate the electric field strength at point B. The electric field strength at point B due to sphere X, E₁=kq₁/r₁² [tex]= (9*10^9Nm^2/C^2) * (72mC) / (8.0m)^2 = 1.01 * 10^6 N/C[/tex] (approx.)
Similarly, the electric field strength at point B due to sphere Y can be calculated as follows, E₂=kq₂/r₂² [tex]= (9*10^9Nm^2/C^2) * (-72mC) / (4.0m)^2 = -4.05 * 10^6 N/C[/tex] (approx.). Therefore, the electric field strength at point A due to the presence of sphere X is [tex]4.05 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex]. The electric field strength at point B due to the presence of sphere X is [tex]1.01 * 10^6 N/C[/tex] and due to the presence of sphere Y is [tex]-4.05 * 10^6 N/C[/tex].
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What is the graph of f(x) = 0.5(4)x (x is an exponent)
It should be a positive graph .
What is the volume of the following triangular prism?
A. 380 m³
B. 398 m³
C. 351 m³
D. 327 m³
Answer:
C-351
Step-by-step explanation:
You just inherited a sum of money from a distant uncle. The only stipulation is that you need to save it for 10 years and then you can do whatever you want with it. The amount of the inheritance is $25,000.
Option one: You can put it into a saving account that earns 6% compounded quarterly.
Option two: You can put it into a checking account that earns 4% compounded monthly.
Option three: You can place it into a money market account that earns 3% compounded daily.
Which option is best for you? Why?
Submit a report detailing the reasons you have for the decision you make.
This flexibility of a money market account makes it an ideal option for people who need to save money without risking their inheritance.
After inheriting $25,000 from a distant uncle, you would like to save it for ten years before doing anything with it. Since you want to save the money, there are several options for keeping it safe and earning interest, including a savings account, a certificate of deposit (CD), and a money market account.
Money market accounts, in my opinion, would be the best place to keep the inheritance. The money market account is a low-risk account with high-interest rates, making it an attractive option for someone who wants to save their money. As a result, it would be reasonable to place the inheritance into a money market account that pays a daily compounded rate of 3%.There are several reasons for choosing this option.
Firstly, the daily compounded interest will generate a higher return over the ten-year period than the simple interest or monthly compounded interest offered by other accounts. Second, the account is FDIC-insured, which means that the account holder is guaranteed to receive their money in the event of a bank failure.
Furthermore, the money market account provides easy access to the account holder's money while still earning interest. Most money market accounts have a limit on the number of withdrawals a person can make per month. Still, the account holder can easily transfer funds into a checking account or withdraw money from an ATM if needed.
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A sixth-grade class recorded the number of letters in each student's first name.
The results are shown in the dot plot.
A dot plot titled lengths of student names show the number of students with a certain number of letters in their name. The data is as follows. 1 dot above 3, 2 dots above 4, 4 dots above 5, 7 dots above 6 and 7, 3 dots above 8, 1 dot above 9, 2 dots above 10, and 3 dots above 11.
Which is the best representation of the center of this data set?
A. 8
B. 5
C. 7
D. 6
Consider a graph of the function y = x² in xy-plane. The minimum distance between point (0, 4) on the y-axis and points on the graph is [1-2] You should rationalize the denominator in the answer. PLEASE HELP ME
The minimum distance between the point (0, 4) on the y-axis and points on the graph is 4.
To find the minimum distance between the point (0, 4) on the y-axis and points on the graph of the function y = x², we can use the concept of perpendicular distance.
The distance between a point (x, y) on the graph and the point (0, 4) is given by the formula:
distance = √((x - 0)² + (y - 4)²) = √(x² + (y - 4)²)
Substituting the function y = x² into the distance formula, we get:
distance = √(x² + (x² - 4)²) = √(x² + (x⁴ - 8x² + 16))
Simplifying further, we have:
distance = √(x⁴ + x² - 8x² + 16) = √(x⁴ - 7x² + 16)
To find the minimum distance, we need to minimize the expression x⁴ - 7x² + 16. Since this is a quadratic-like expression, we can use calculus to find the minimum.
Taking the derivative of x⁴ - 7x² + 16 with respect to x, we get:
d/dx (x⁴ - 7x² + 16) = 4x³ - 14x
Setting the derivative equal to zero to find critical points:
4x³ - 14x = 0
Factorizing, we have:
2x(2x² - 7) = 0
This gives us two critical points: x = 0 and x = ±√(7/2).
Next, we evaluate the expression x⁴ - 7x² + 16 at these critical points and the endpoints of the interval:
f(0) = 0⁴ - 7(0)² + 16 = 16
f(±√(7/2)) = (√(7/2))⁴ - 7(√(7/2))² + 16 ≈ 4.157
Comparing these values, we find that the minimum distance occurs at x = 0, giving us a minimum distance of √(0⁴ - 7(0)² + 16) = √16 = 4.
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Please answer ASAP I will brainlist
Answer:
A) The y-intercept(s) is/are 2
Step-by-step explanation:
Y-intercepts are where the graph of a function cross over the y-axis. In this case, the line passes through y=2, which is the y-intercept.
In December 2016 the average price of unleaded
Which exponential equation is e
quivalent to the logarithmic equation below? log=200 a
A. 200 = 10
B. 200¹0 = a
C. a¹0 = 200
D. 10 = 200 SUBMIT
The exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.
Which rule of logarithms should we use here?The rule of logarithms that we should use here is given below:
[tex]\log \text{x} = \text{a} \iff 10^{\text{a}} = \text{x}[/tex]
We can find the equivalent exponential equation below:The given expression is Log = 200 a.
We can follow the rule log x = a ⇔ 10^a = x to convert this logarithmic equation to an exponential one.
Log = 200 a can be rewritten as a¹⁰ = 200
Therefore, we have found that the exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.
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Statement: Reasons:
DF=EG.
DE
Prove: DE=FG
Statements
DF=EG
DF=DE+EF.
EG=EF+FG
DE+EF=EF+ FG
FL
G
Answer:
az
Step-by-step explanation:
6 + 9x^2 + 3x^3 + 2x^4 - 12x
how many positive, negative, and complex zeros are there?
There are no positive or negative zeros, and the number of complex zeros cannot be determined without further information.
To determine the number of positive, negative, and complex zeros of the given polynomial [tex]6 + 9x^2 + 3x^3 + 2x^4 - 12x,[/tex] we need to analyze its behavior and apply the properties of polynomial functions.
Positive Zeros:
Positive zeros are the values of x for which the polynomial evaluates to zero.
To find positive zeros, we set the polynomial equal to zero and solve for x.
However, in this case, we can see that all the coefficients of the terms in the polynomial are positive.
Therefore, there are no positive zeros.
Negative Zeros:
Negative zeros are the values of x for which the polynomial evaluates to zero.
Similar to positive zeros, we set the polynomial equal to zero and solve for x.
However, in this case, we can see that all the coefficients of the terms in the polynomial are positive.
Therefore, there are no negative zeros.
Complex Zeros:
Complex zeros occur when the polynomial has complex roots. Since the given polynomial has only real coefficients, complex zeros will occur in conjugate pairs.
To determine the number of complex zeros, we need to examine the degree of the polynomial.
In this case, the highest power of x is [tex]4 (x^4),[/tex] indicating a fourth-degree polynomial.
A fourth-degree polynomial can have at most four complex zeros. However, we cannot determine the exact number of complex zeros without further information or solving the polynomial explicitly.
In conclusion, the given polynomial has no positive or negative zeros due to all coefficients being positive.
The number of complex zeros cannot be determined without additional information.
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You own a portfolio that has $3,000 invested in Stock A and $4,100 invested in Stock B. Assume the expected returns on these stocks are 10 percent and 16 percent, respectively. What is the expected return on the portfolio?
The expected return on the portfolio is approximately 13.465%.
To calculate the expected return on the portfolio, we need to consider the weights of each stock in the portfolio.
Let's denote the weight of Stock A as wA and the weight of Stock B as wB. The weight of a stock is the proportion of the total portfolio value that is invested in that stock.
Given that $3,000 is invested in Stock A and $4,100 is invested in Stock B, we can calculate the weights as follows:
wA = $3,000 / ($3,000 + $4,100) = $3,000 / $7,100
wB = $4,100 / ($3,000 + $4,100) = $4,100 / $7,100
Next, we need to calculate the weighted average of the expected returns of the two stocks using their respective weights:
Expected return on the portfolio = (wA * Return on Stock A) + (wB * Return on Stock B)
Expected return on the portfolio = (wA * 10%) + (wB * 16%)
Substituting the calculated weights into the equation:
Expected return on the portfolio = ($3,000 / $7,100 * 10%) + ($4,100 / $7,100 * 16%)
Simplifying the equation:
Expected return on the portfolio = (0.4225 * 10%) + (0.5775 * 16%)
Expected return on the portfolio = 0.04225 + 0.0924
Expected return on the portfolio = 0.13465 or 13.465%
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Determine the surface area and volume Note: The base is a square.
The surface area and volume of the square pyramid is 96 squared centimeter and 48 cubic centimeters respectively.
What is the surface area and volume of the square pyramid?The surface area of a square pyramid is expressed as:
SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]
The volume of a square pyramid is expressed as:
Volume = [tex]a^2*\frac{h}{3}[/tex]
Where a is the base edge and h is the height.
From the figure a = 6cm
First, we determine the h, using pythagorean theorem:
h² = 5² - (6/2)²
h² = 5² - 3²
h² = 25 - 9
h² = 16
h = √16
h = 4 cm
Solving for surface area:
SA = [tex]a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }[/tex]
[tex]= a^2 + 2a \sqrt{\frac{a^2}{4}+h^2 }\\\\= 6^2 + 2*6 \sqrt{\frac{6^2}{4}+4^2 }\\\\= 36 + 12 \sqrt{\frac{36}{4}+16 }\\\\= 36 + 12 (5)\\\\= 36 + 60\\\\= 96 cm^2[/tex]
Solving for the volume:
Volume = [tex]a^2*\frac{h}{3}[/tex]
[tex]= a^2*\frac{h}{3}\\\\= 6^2*\frac{4}{3}\\\\= 36*\frac{4}{3}\\\\=\frac{144}{3}\\\\= 48 cm^3[/tex]
Therefore, the volume is 48 cubic centimeters.
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25. Use the relationship in the diagrams below to solve for the given variable.
Justify your solution with a definition or theorem.
Answer:
x = 110°
Step-by-step explanation:
The opposite angles are equal in a parallelogram
3x - 60 = 2x + 50
⇒ 3x - 2x = 60 + 50
⇒ x = 110°
Answer:
x = 110°
Step-by-step explanation:
As the top and bottom line segments of the given shape are the same length and parallel (indicated by the tick marks and arrows), the shape is a parallelogram.
As the opposite angles of a parallelogram are equal, to find the value of the variable x, equate the two angle expressions and solve for x:
[tex]\begin{aligned}3x-60^{\circ}&=2x+50^{\circ}\\3x-60^{\circ}-2x&=2x+50^{\circ}-2x\\x-60^{\circ}&=50^{\circ}\\x-60^{\circ}+60^{\circ}&=50^{\circ}+60^{\circ}\\x&=110^{\circ}\end{aligned}[/tex]
Therefore, the value of x is 110°.
Note: There must be an error in the question. If x = 110°, each angle measures 270°, which is impossible since the sum of the interior angles of a quadrilateral is 360°.
21. An RSTU rectangle is drawn on the coordinate plane with coordinates R(-1, 5), S(4, 5), T(4, 9) and then translated by T(2,-3), then the image coordinates of point U are
The image coordinates of point U, after translating the RSTU rectangle by T(2,-3), would be U(6, 6).
To find the image coordinates of U, we need to apply the translation vector T(2,-3) to each of the original coordinates.
The translation vector represents the horizontal and vertical distances by which each point is moved.
Starting with the original coordinates of point U, which are (4, 9), we add the horizontal distance of 2 to the x-coordinate and subtract the vertical distance of 3 from the y-coordinate.
Therefore, the new x-coordinate of U is 4 + 2 = 6, and the new y-coordinate is 9 - 3 = 6.
Thus, the image coordinates of point U after the translation are (6, 6). This means that U has been moved 2 units to the right and 3 units downward from its original position.
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Please help! :')
Prove that the two circles shown below are similar. (10 points)
Circle X is shown with a center at negative 2, 8 and a radius of 6. Circle Y is shown with a center of 4, 2 and a radius of 3.
how do you find out how many positive zeros and negative zeros are in a polynomial based on a graph?
In this context, the term "zeros" refers to the roots or x-intercepts.
You're looking for where the graph either,
a) touches the x axis, or,b) crosses the x axisSee the diagram below. That example shows two positive roots, because each root is to the right of the vertical y axis.
What is the distance between points R (5, 7) and S(-2,3)?
Answer:
d ≈ 8.1
Step-by-step explanation:
calculate the distance d using the distance formula
d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]
with (x₁, y₁ ) = R (5, 7 ) and (x₂, y₂ ) = S (- 2, 3 )
d = [tex]\sqrt{(-2-5)^2+(3-7)^2}[/tex]
= [tex]\sqrt{(-7)^2+(-4)^2}[/tex]
= [tex]\sqrt{49+16}[/tex]
= [tex]\sqrt{65}[/tex]
≈ 8.1 ( to 1 decimal place )
help please its due in 50 minutes ill mark brainliest answer too and no need to show work
The output value f(3) in the functions f( x ) = 3x + 5, f( x ) = [tex]\frac{1}{2}x^2-1.5[/tex] and f( x ) = [tex]\frac{3}{2x}[/tex] is 14, 3 and 1/2 respectively.
What is the output value of f(3) in the given functions?Given the functions in the question:
f( x ) = 3x + 5
f( x ) = [tex]\frac{1}{2}x^2-1.5[/tex]
f( x ) = [tex]\frac{3}{2x}[/tex]
To evaluate each function at f(3), we simply replace the variable x with 3 and simplify.
a)
f( x ) = 3x + 5
Replace x with 3:
f( 3 ) = 3(3) + 5
f( 3 ) = 9 + 5
f( 3 ) = 14
b)
f( x ) = [tex]\frac{1}{2}x^2-1.5[/tex]
Replace x with 3:
[tex]f(3) = \frac{1}{2}(3)^2 - 1.5\\\\f(3) = \frac{1}{2}(9) - 1.5\\\\f(3) = 4.5 - 1.5\\\\f(3) = 3[/tex]
b)
f( x ) = [tex]\frac{3}{2x}[/tex]
Replace x with 3:
[tex]f(3) = \frac{3}{2(3)} \\\\f(3) = \frac{3}{6} \\\\f(3) = \frac{1}{2}[/tex]
Therefore, the output value of f(3) is 1/2.
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Please help with 23a and 24a
Answer:
perimeter = 75 cm , area ≈ 28.3 in²
Step-by-step explanation:
23 (a)
since the figure is being enlarged by a scale factor of 5
then the perimeter is increased by a factor of 5.
perimeter of larger shape = 5 × 15 = 75 cm
24
the area (A) of the sector is calculated as
A = area of circle × fraction of circle
= πr² × [tex]\frac{90}{360}[/tex]
= π × 6² × [tex]\frac{1}{4}[/tex]
= 36π × [tex]\frac{1}{4}[/tex]
= 9π
≈ 28.3 in² ( to 1 decimal place )
Question #5
Find the measure of the indicated arc.
OOOO
90°
80°
100°
70°
G
H
40°
F
The value of the required arc in the figure is solved to be
80°How to find the value of the arcThe inscribed angle is given in the problem as 40 degrees. This is the angle formed at the circumference of the circle
The relationship between inscribed angle and the intercepted arc is
intercepted arc = 2 * inscribed angle
in the problem, we have that
intercepted arc = ?
inscribed angle = 40
plugging in the values
intercepted arc = 2 * 40
intercepted arc = 80 degrees
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Use the parallelogram ABCD to find the following.
8. part 2
b. AE=
d. m<DCB=
f. m<ADC=
Answer:
b. 7
d. 120°
f. 60°
Step-by-step explanation:
The properties of a parallelogram are:
Opposite sides are parallel and congruent.Opposite angles are equal.Adjacent angles are supplementary.The diagonals bisect each other.The sum of the interior angles is 360 degrees.For Question:
b.
AE=CE=7 since diagonals of parallelogram bisect each other
d.
m ∡ DCB= m ∡DAB=120° Opposite angle in a parallelogram is congruent or equal.
f.
m ∡ ADC=?
Here
m ∡ADC+ m ∡DAB=180° being co interior angle
m ∡ADC+120°=180°
m ∡ADC=180°-120°=60°
m ∡ADC=60°
Answer:
b) AE = 7
d) m∠DCB = 120°
f) m∠ADC = 60°
Step-by-step explanation:
Part bThe diagonals of a parallelogram always bisect each other.
Therefore, point E (the point of intersection of the two diagonals) is the midpoint of diagonal AC. So AE = EC.
As EC = 7, then AE = 7.
[tex]\hrulefill[/tex]
Part dAs the measure of the opposite angles of a parallelogram are equal, then m∠DCB is equal to m∠DAB. From inspection of the given parallelogram, we can see that m∠DAB = 120°. Therefore:
m∠DCB = 120°
[tex]\hrulefill[/tex]
Part fAdjacent angles of a parallelogram are supplementary (sum to 180°).
Angle DAB and angle ADC are adjacent angles, so their sum is 180°.
Therefore:
⇒ m∠ADC + m∠DAB = 180°
⇒ m∠ADC + 120° = 180°
⇒ m∠ADC + 120° - 120° = 180° - 120°
⇒ m∠ADC = 60°
Jalen bought a new iPad. The screen has a perimeter of inches 36 inches and an area of 80 square inches. What are the dimensions of the iPad’s screen?
Let's assume the dimensions of the iPad's screen are represented by length (L) and width (W). We can use the given information to set up two equations.
Perimeter equation:
The perimeter of a rectangle is given by the formula: 2(L + W). In this case, the perimeter is given as 36 inches.
So, 2(L + W) = 36.
Area equation:
The area of a rectangle is given by the formula: L * W. In this case, the area is given as 80 square inches.
So, L * W = 80.
We now have a system of two equations:
2(L + W) = 36
L * W = 80
From equation 1, we can simplify it to L + W = 18 and rewrite it as L = 18 - W.
Now substitute this value of L into equation 2:
(18 - W) * W = 80
Expanding the equation:
18W - W^2 = 80
Rearranging the equation to quadratic form:
W^2 - 18W + 80 = 0
Factoring or using the quadratic formula, we find the two possible values for W.
The solutions are W = 8 and W = 10.
Now we can substitute these values back into the equation L = 18 - W to find the corresponding values of L.
For W = 8:
L = 18 - 8 = 10
For W = 10:
L = 18 - 10 = 8
Therefore, the dimensions of the iPad's screen can be either 8 inches by 10 inches or 10 inches by 8 inches.
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Todd noticed that the gym he runs seems less crowded during the summer. He decided to look at customer data to see if his impression was correct.
Week
5/27 to 6/2
6/3 to 6/9
6/10 to 6/16
6/17 to 6/23
6/24 to 6/30
7/1 to 7/7
Use
618 people
624 people
618 people
600 people
570 people
528 people
A: What is the quadratic equation that models this data? Write the equation in vertex form.
B: Use your model to predict how many people Todd should expect at his gym during the week of July 15.
Todd should expect_______people.
Todd should expect approximately 624 people at his gym during the week of July 15.
A: To find the quadratic equation that models the data, we can use the vertex form of a quadratic equation:
[tex]y = a(x - h)^2 + k[/tex] where (h, k) represents the vertex of the parabola.
Let's analyze the data to determine the vertex. We observe that the number of people is highest during the first week and gradually decreases over the following weeks.
This suggests a downward-opening parabola.
From the data, the highest point occurs during the week of 6/3 to 6/9 with 624 people.
Therefore, the vertex is located at (6/3 to 6/9, 624).
Using the vertex form, we have:
[tex]y = a(x - 6/3 to 6/9)^2 + 624[/tex]
Now, we need to find the value of 'a.'
To do this, we can substitute any other point and solve for 'a.' Let's use the data from the week of 5/27 to 6/2:
[tex]618 = a(5/27 to 6/2 - 6/3 to 6/9)^2 + 624[/tex]
Simplifying the equation and solving for 'a,' we find:
[tex]618 - 624 = a(-6/3)^2[/tex]
-6 = 4a
a = -3/2
Therefore, the quadratic equation in vertex form that models the data is:
[tex]y = (-3/2)(x - 6/3 to 6/9)^2 + 624[/tex]
B: To predict the number of people Todd should expect during the week of July 15, we substitute x = 7/15 into the equation and solve for y:
[tex]y = (-3/2)(7/15 - 6/3 to 6/9)^2 + 624[/tex]
Simplifying the equation, we find:
[tex]y = (-3/2)(1/15)^2 + 624[/tex]
y = (-3/2)(1/225) + 624
y = -3/450 + 624
y = -1/150 + 624
y = 623.993
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HELP ME PLS I'LL MARK BRAINLIEST AND GIVE U 13 POINTS
Answer:
The answer is number 3
Y =-3/7X + 3
Step-by-step explanation:
Substitue with the value of the two points in all answers
The value of the left side must equal the value or the right side
For instance,
The two points are (0,3) & (7,0)
Substitue in the first answer with the point (0,3)
3 = - 3 (rejected)
Second answer
3 = 3 works for the point (0,3) then also Substitue with the other point (7,0)
0 = 6 (rejected)
The third answersub. With point (0,3).
3 = 3 it worksSub. With point (7,0)
0= - 3+3 0=0 it worksThen that's the right one
Circle 1 is centered at (−4,−2) and has a radius of 3 centimeters. Circle 2 is centered at (5,3) and has a radius of 6 centimeters.
What transformations can be applied to Circle 1 to prove that the circles are similar?
Enter your answers in the boxes.
The circles are similar because you can translate Circle 1 using the transformation rule ( , ) and then dilate it using a scale factor of .
The circles are similar because you can translate Circle 1 using the transformation rule (9, 5) and then dilate it using a scale factor of 2.
To prove that Circle 1 and Circle 2 are similar, we need to identify the transformations that can be applied to Circle 1 to obtain Circle 2.
First, let's consider the translation of Circle 1. The translation rule is given by (a, b), where a represents the horizontal shift and b represents the vertical shift.
In this case, to translate Circle 1 to align with Circle 2, we need to shift it 9 units to the right and 5 units up. Therefore, the translation rule for Circle 1 is (9, 5).
Next, let's consider the dilation. A dilation is a transformation that changes the size of the figure but preserves its shape. The scale factor, denoted by k, determines the amount of scaling. In this case, Circle 1 needs to be dilated to match the size of Circle 2.
The scale factor can be determined by comparing the radii of the two circles. The radius of Circle 1 is 3 centimeters, while the radius of Circle 2 is 6 centimeters. The scale factor is obtained by dividing the radius of Circle 2 by the radius of Circle 1: 6/3 = 2.
Therefore, the transformation applied to Circle 1 to prove that the circles are similar is a translation by (9, 5) followed by a dilation with a scale factor of 2.
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10. Consider the quadratic function f(x)=x² +6x. Solve the inequality for f(x) > -5.
Answer: x<-5
x>-1
The solution to the inequality f(x) > -5 is x < -5 or x > -1.
To solve the inequality f(x) > -5 for the quadratic function f(x) = x^2 + 6x, we need to find the values of x that satisfy the inequality.
First, set up the inequality:
x^2 + 6x > -5
Next, move all terms to one side of the inequality to get a quadratic expression:
x^2 + 6x + 5 > 0
To solve this quadratic inequality, we can factor it:
(x + 5)(x + 1) > 0
Now, we need to determine the sign of the expression for different intervals on the x-axis.
a) When x < -5:
If x is less than -5, both (x + 5) and (x + 1) are negative, so their product is positive.
Thus, the inequality is satisfied for x < -5.
b) When -5 < x < -1:
If x is between -5 and -1, (x + 5) is positive, but (x + 1) is negative. The product of a positive and a negative number is negative.
Thus, the inequality is not satisfied for -5 < x < -1.
c) When x > -1:
If x is greater than -1, both (x + 5) and (x + 1) are positive, so their product is positive.
Thus, the inequality is satisfied for x > -1.
Therefore, x -5 or x > -1 is the answer to the inequality f(x) > -5.
In summary:
x < -5 or x > -1.
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