Answer:
43.75 ft²
Step-by-step explanation:
Let r = the radius of the semicircle
and h = the height of the rectangle
Then 2r = the width of the window
The formula for the perimeter of a circle is C = 2πr,
so, πr = the perimeter of the semicircle
The perimeter of the window is
P = πr + 2h + 2r = 25
2h + (π +2)r = 25
h = ½[25 - (π + 2)r]
(1) h = 12.5 - (π/2 +1)r
The formula for the area of a circle is A= πr², so
½πr² = the perimeter of the semicircle
The area of the window is
(2) A = ½πr² + 2rh
Substitute (1) into (2).
A = ½πr² + 2r[12.5 - (π/2 +1)r] = ½πr² + 25r - (π +2)r²
A = 25r - (π + 2 - π/2)r²
(3) A = -(π/2 + 2)r² + 25r
This is the equation for a downward opening parabola.
One way to find the vertex is to set the first derivative equal to zero.
dA/dr = -2(π/2 + 2)r + 25 = 0
-(π + 4)r + 25 = 0
-(π + 4)r = -25
r = 25/(π + 4)
(4) r ≈ 3.50 ft
The maximum area occurs when r = 3.50 ft.
Substitute (4) into (1).
h = 12.5 - (π/2 +1)(3.50) = 12.5 - (2.571× 3.50) = 12.5 - 9.00 = 3.50
(4) h = 3.50 ft
Substitute (4) into (2)
A = 1.571(3.50)² + 2×3.50×3.50 = 19.25 + 24.50
A = 43.75 ft²
The area of the largest possible Norman window with a perimeter of 25 ft is 43.75 ft².
The maximum area of a Norman window with a given perimeter of 25 feet can be found by creating an equation for the area, taking its derivative, setting it equal to zero and solving for the window's dimensions. This involves calculus, namely the method for optimization problems.
Explanation:The problem involves maximizing the area of a Norman window given a certain perimeter. The Norman window is composed of a rectangle and a semicircle, where the diameter of the semicircle equals the width of the rectangle. First, let's denote the width of the rectangle or the diameter of the semicircle as x. The radius of the semicircle will then be x/2. The height of the rectangle can be represented as 25 - x (since the perimeter of the window should equal 25 feet).
The area of a rectangle is height multiplied by width. The area of a semicircle is (1/2)πr2. So to find the area A of the Norman window, we add the area of the rectangle and the semicircle: A = x(25-x) + (1/2)π*(x/2)2.
To find the maximum area, we need to take the derivative of A with respect to x, set it equal to zero, and solve for x. We find x approximates to around 7.64 feet (after using calculus concepts). Then substitute x = 7.64 into the area function A and solve for A, which gives us the maximum area of the Norman window.
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The bus from Karen's house to school runs 26 to 75 minutes of the bus runs on Monday at 7:05am at which time could the bus run next
I’m pretty sure the answer is 8:25 a.m
Find the product of x^2+2x-4 and 3x.
a. 3x^3+6x^2+12x
b. 3x^3+6x^2-12x
c. 3x^3+2x-4
d. -3x^3
[tex] 3x({x}^{2} + 2x - 4)[/tex]
[tex]3x \times {x}^{2} = {3x}^{3} [/tex]
[tex]3x \times 2x = 6 {x}^{2} [/tex]
[tex]3x \times - 4 = - 12x[/tex]
[tex]b. \: 3 {x}^{3} + 6 {x}^{2} - 12x[/tex]
PLEASE HELP! URENT!
What is the measure of ∠R ?
Enter your answer as a decimal in the box. Round only your final answer to the nearest hundredth.
m∠R=
°
Answer:
R = 48.59 degrees
Step-by-step explanation:
We know that
sin theta = opposite / hypotenuse
sin R = 30/40
sin R = 3/4
Taking the inverse sin of each side
sin^-1 (sin R) = sin ^-1(3/4)
R = 48.59037789
To the nearest hundredth
R = 48.59 degrees
Lines p and q are perpendicullllar if the slope of line p is 2 what is the slope of line q
q= -1/2
perpendicular slopes are those that are opposite sign and reciprocal of the original given slope
The point (4, -2) lies on a circle and the center of the circle is at the point (4, 1).
What is the equation of this circle?
A. (x - 1)2 + (y - 4)2 = 9
B. (x - 1)2 + (y - 4)2 = 3
C. (x + 4)2 + (y + 1)2 = 9
D. (x - 4)2 + (y - 1)2 = 9
Answer:
D
Step-by-step explanation:
The equation of the circle is (x-x0)^2 + (y-y0)^2 = r^2, where (x0,y0) is the center of the circle. Radius is the distance from any point to the center. Using distance formula you get r = 3.
The equation of the circle having center at point (4,1) and passing through the point (4,-2) is Option (D) [tex](x - 4)^{2} + (y - 1)^{2} = 9[/tex]
Equation of circle in parametric form -A circle having radius r and having center at (x0,y0) can be represented by the parametric equation as -
[tex](x - x0)^{2} + (y - y0)^{2} = r^{2}[/tex]
Any point say (x1,y1) which lies on the circle, must satisfy the given parametric equation of the circle.
How to frame the equation of circle from the information given in the question ?Given that the point (4,-2) lies on the circle. Also the center of circle is at (4,1). Therefore, calculating the radius of the circle which is the distance from the center to the point lying on the circle.
Using distance formula,
radius = [tex]\sqrt{(4 - 4)^{2} + (-2 - 1)^{2} }[/tex] = [tex]\sqrt{9 }[/tex] = [tex]3[/tex]
Thus forming the equation of circle from the general parametric equation,
⇒ [tex](x - 4)^{2} + (y - 1)^{2} = 3^{2}[/tex]
∴ [tex](x - 4)^{2} + (y - 1)^{2} = 9[/tex]
Therefore the equation of circle is Option (D) [tex](x - 4)^{2} + (y - 1)^{2} = 9[/tex]
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Given the following sets, which of the following is false? A={4, 6, 8} B={-2, 0, 2, 4} C={13, 23, 33, 43}
Answer:
gfjhgfterjrjtryuyrt
Step-by-step explanation:
hgfdjjrfdsfdjuytitdfgfdsuygeryauvfudsgygwerauyerwaguyfdsguyagsuydzgfuygewsabfuyguyeravcuyeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraeraerajhbfcds
A car of the same type and make is 3 1/2 years old. Estimate the value of the car.
The values in the scatter diagram indicate;
a) The graph of the data showing the location of the car that costs £2600 and is 4 years of age created with MS Excel is attached
b) The value of the cars decreases as the age increases
c) £3687.9
To address the question, plot the new car data on a scatter diagram, describe the trend of decreasing value with increasing age, and estimate the value of a car based on its age by interpolating between data points.
The question requests steps on how to represent data on a scatter diagram as well as how to interpret and estimate values from it.
a. Add the new point to the scatter diagram for a car that's 4 years old and valued at £2600. This point (4, £2600) should be plotted on the graph along with the other data points.
b. Describe the relationship by observing how the value tends to change as the age of the cars increases. Generally, as cars age, their value decreases, which suggests a negative relationship between the age and the value of the cars.
c. For estimating the value of a car that is 3.5 years old can be found as follows; The equation of the trend line for the scatter diagram obtained usinfg MS Excel is; y = -1290.2·x + 8203.6, indicates that when x = 4, we get;
y = -1290.2 × 3.5 + 8203.6
-1290.2 × 3.5 + 8203.6 =
The estimated price of the car is £3687.9
The complete question found through search is presented as follows;
The scatter diagram shows information about 10 cars.
The 10 cars are of the same type and make
Another car is 4 years old and has been valued at £2,600
a) Show this information on the graph
b) Describe the relationship between the age and the values of the cars
A car of the same type and age is 3¹/₂ years old
c) Estimate the value of the car
One part of a circle is shaded 1/3 and another is shaded 1/2. How much of the circle is shaded? Write your answer as a fraction in simplest form.
Answer:
5/6
Step-by-step explanation:
1/3 + 1/2 = 5/6
Brian and Carlos are sharing a large sub sandwich. Brian ate 2 / 5 of the sandwich, and Carlos ate 1 / 5 of the sandwich. How much of the sandwich is left over?
if I'm right it would be 2/5. you get your answer by adding 2/5 and 1/5 then subtract 5/5- 3/5 to get 2/5
what quadrilateral always has 4 congruent angles and opposite sides that are congruent and parallel?
Answer:
See below.
Step-by-step explanation:
A rectangle has 4 * 90 degree angles and opposite sides are congruent and parallel. A square also fits the bill.
Angenita drives to work at 50 mph and arrives 1 minute late. she drives to work at 60 mph and arrives 5 minutes early.
Answer:
2 mpm
Step-by-step explanation
The sum of 5/11 and 4/11 is
equal to 1
less than 1
greater than 1
Train a travels in 840 miles in 7.5 hours. Train b travels 1080 miles in 9 hours. Which train was traveling at the fastest speed, and what was that speed?
Answer:
Train B is traveling faster at a speed of 120 MPH
How to solve:
Train A is 840/7.5 = 112 which is the MPH so its 112 MPH
Train B is 1080/9 = 120 MPH
Final answer:
Train B is the faster train with an average speed of 120 miles per hour, compared to Train A which has an average speed of 112 miles per hour.
Explanation:
The question is asking to compare the average speed of two trains. To find the average speed, we divide the total distance traveled by the time taken. For Train A, it travels 840 miles in 7.5 hours, and for Train B, it travels 1080 miles in 9 hours.
Calculating the average speed of Train A: 840 miles / 7.5 hours = 112 miles per hour.
Calculating the average speed of Train B: 1080 miles / 9 hours = 120 miles per hour.
Therefore, Train B is traveling at the fastest speed with an average of 120 miles per hour.
You are buying shoes online.The selling price is $29.99. Round to the nearest penny if necessary.
it's 30 dollars because it's the closest whole number
Final answer:
To calculate the total cost of an online purchase, round the sales tax to the nearest penny and add it to the item's price. When combining multiple items, sum their prices and add the rounded sales tax to find the total cost.
Explanation:
When buying shoes or any item online and encountering a selling price that includes cents, you may need to perform rounding if further calculations are required. For example, to round $6.375 to the nearest penny, we would get $6.38 because the third digit after the decimal is 5 or more, thus we increase the second digit by one. From there, we would add the rounded sales tax to the original price of the item to find the total cost.
In a different scenario where you are adding two prices together, like $8.99 and $11.99, you get the subtotal of $20.98 before tax. Then calculate the sales tax by converting the percent to a decimal and multiplying by the subtotal, rounding again to the nearest cent if necessary. Finally, add the sales tax to the subtotal to get the total cost. For instance, if the sales tax was $1.47, adding this to the subtotal of $20.98 would give you a total of $22.45.
Solve the equation by using tu basic properties of logarithms log(2x)=3 (Picture provided)
Answer:
Option b
Step-by-step explanation:
According to the property of sum of logarithms we know that
[tex]log(ab) = log(a) + log(b)[/tex].
In this case we have the equation:
[tex]log(2x) = 3[/tex]
Using the property of sum of logarithms:
[tex]log(2) + log(x) = 3[/tex]
[tex]log(x) = 3 - log(2)[/tex]
We also know that:
[tex]10 ^{(logx)} = x[/tex] -------- Inverse logarithm
So:
[tex]x = 10 ^{3-log(2)}[/tex]
[tex]x = 500[/tex]
Another easiest way to solve it is the following:
Make [tex]w = 2x[/tex].
Then:
[tex]log(2x) = log(w)[/tex]
[tex]log(w) = 3[/tex]
[tex]w = 10^3[/tex] -------- Inverse logarithm property
[tex]w = 1000[/tex]
but [tex]w= 2x[/tex]. Then:
[tex]2x = 1000\\\\x = 500[/tex]
Find the total area of the solid figure.
60 sq. in.
78 sq. in.
90 sq. in.
Answer:
245.04
Step-by-step explanation:
The equation to a cylinder for area is A=2πrh+2πr^2
Fill in and solve.
A=2(3.14)(3)(10)+2(3.14)(3)^2
So using PEMDAS you get the answer 245.04
While taking @Jtakes700's answer and putting it into the form your asking, or putting into one of your options, you would get 78pi sq. in.
Question 2(Multiple Choice Worth 4 points) (08.02)A pair of equations is shown below. x + y = 5 y = one halfx + 2 If the two equations are graphed, at what point do the lines representing the two equations intersect? (2, 5) (5, 2) (2, 3) (3, 2)
Answer:
[tex](2,3)[/tex]
Step-by-step explanation:
The first equation is [tex]x+y=5[/tex]
The second equation is [tex]y=\frac{1}{2}x+2[/tex]
When we graph these two equations, they will meet at a point which represent the solution of the two equations.
We can solve the two equations simultaneously to determine their point of intersection.
Let us substitute the second equation into the first equation to get;
[tex]x+\frac{1}{2}x+2=5[/tex]
Multiply through by 2 to get;
[tex]2x+x+4=10[/tex]
Group similar terms to obtain;
[tex]2x+x=10-4[/tex]
Simplify;
[tex]3x=6[/tex]
Divide both sides by 3;
[tex]\Rightarrow x=2[/tex]
Put [tex]x=2[/tex] into the second equation;
[tex]y=\frac{1}{2}(2)+2[/tex]
[tex]\Rightarrow y=1+2[/tex]
[tex]\Rightarrow y=3[/tex]
Therefore the graphs of the two functions intersect at (2,3)
See graph in attachment.
Name the different forms of linear equations you learned. If (1, 2) and
(3, 4) lie on the same line, give the slope-intercept equation of the line.
2
Explain how to find the inverse of f(x) = (x-1)/4
2
What is average rate of change? How do you calculate it? What can you tell about the average rate of change for linear functions? Compare the slopes of horizontal and vertical lines.
3/4
Compare the processes of graphing linear equations and inequalities with examples of your own.
3/4
Create a real-life scenario in which you show how you interpret key features of functions and graphs such as slope and y-intercept?
Module 2
1
What is the definition of “i”? What do we use it for? Demonstrate with an example of your own.
2
Why are radicals simplified before adding and subtracting? Explain your reasoning by adding sqrt8 and sqrt32. Compare the process to multiplying and dividing.
2
Give an example of an expression with a rational exponent and explain how to convert it into radical form.
3/4
A good practice in mathematics is to always check your work. Explain why it is very important to do so when you are solving equations that involve radical expressions?
Where in real life do you work with radical expressions and functions? Give an example of your own.
Module 3
Webbs DOK
Discussion Prompt
1
How many different ways can you solve a quadratic equation? List them.
2
Create a trinomial that can be factored and write it in standard form.
2
Factor x^2 – 7x + 10, 4x^2 – 81 and p(x)=3x^3-12x
3/4
What key features of a quadratic graph can be identified and how are the graphs affected when constants or coefficients are added to the parent quadratic equations? Compare the translations to the graph of linear function. Create examples of your own to explain the differences and similarities.
3/4
A ball is kicked into the air and follows the path described by
h(t)= -4.9t^2+6t+0.6, where t is the time in seconds and h is the height in meters above the ground. Find the maximum height of the ball. What value would you have to change in the equation if the maximum height of the ball is more than 2.4 meters?
The solution to this exercise has been attached below. The problem has been solved in this way:
________________
1. Different forms of linear equations. Point slope-intercept equation of the line that passes through two points.2. Inverse Function. 3. Average Rate of Change.4. Comparison of linear equations and inequalities5. Real-life problems6. Imaginary Number7. Radicals8. Rational exponent and radical form9. Radical expressions10. Quadratic equation11. Trinomial12. Factoring expressions13. Quadratic and linear graph14. A problem of heightAnswer:
Everything he said was correct!
Step-by-step explanation:
I hope this helps!
- sincerelynini
When you multiply expressions with the same base, you ______ the exponents. Add multiply
Answer:
... you add the exponents
Step-by-step explanation:
An exponent is a way of showing repeated multiplication. For example, ...
x^2 means x·x
and
x^3 means x·x·x
If we multiply these two expressions together, we get ...
(x^2)·(x^3) = (x·x)·(x·x·x) = x·x·x·x·x
This is a product in which x appears as a factor 5 times, so we can write it using an exponent as ...
x·x·x·x·x = x^5
___
That is, when we multiply the expressions, we add the numbers of times the factor appears in the product:
2 + 3 = 5
x^2 · x^3 = x^(2+3) = x^5
Answer: Add
Step-by-step explanation: When you multiply expression with the same base you ADD the exponents. The exponent tells you how many times you multiply the base by itself. This means the exponent corresponds to the amount of base replications. If you add the exponents you are adding the replications of the base.
Use the x-intercept method to find all real solutions of the equation.
x^3-7x^2+4x+12=0
Answer:
the answer is c for apex
Step-by-step explanation:
Answer:
a. [tex]x=-1,2,\:or\:6[/tex]
Step-by-step explanation:
The given equation is
[tex]x^3-7x^2+4x+12=0[/tex]
To find all real solutions using the x-intercept method, we to graph the corresponding function using a graphing tool.
The corresponding function is;
[tex]f(x)=x^3-7x^2+4x+12[/tex]
The real solutions to [tex]x^3-7x^2+4x+12=0[/tex], are the x-intercepts of the graph of the corresponding function.
From the graph the x-intercepts are
(-1,0),(2,0) and (6,0).
Therefore the real solutions are
[tex]x=-1,2,\:or\:6[/tex]
HELP PLEASE!!! NEED FAST ANWER!!
Identify the slope of a line that described by:
down 3 and right 5
m=3/5
m=-3/5
m=5/3
m=-5/3
which one is it???
Answer: second option.
Step-by-step explanation:
The equation of the line in slope-intercept form is:
[tex]y=mx+b[/tex]
Where m is the slope of the line and b the y-intercept.
We know that we can calculate the slope with:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Therefore, keeping the above on mind, you have that:
- If the line moves upward from the left to right, then the slope is positive.
- If the line moves down from the left to right, then the slope is negative.
Then, if the slope described moves 3 units down and 5 units to the right, the it is negative. Therefore:
[tex]m=-\frac{3}{5}[/tex]
Answer:
[tex]-\frac{3}{5}[/tex]
Step-by-step explanation:
We are to identify the slope of a line which is described as:
'down 3 and right 5'
We know that a straight line which is sloping downwards from left to right has a negative slope.
And a straight line which goes upwards from left to right side has a positive slope.
According to the describe slope of a line, it goes 3 units down and 5 units right so it makes a negative slope which is equal to [tex]-\frac{3}{5}[/tex].
Find the area of the trapezoid. leave your answer in simplest radical form.
Answer:
Area[tex]=70in^2[/tex]
Step-by-step explanation:
The given diagram is a tra-pezoid.
The area of a tra-pezoid is calculated using the formula;
Area[tex]=\frac{1}{2} (Sum\:of\:parallel\:sides)\times height[/tex]
This implies that;
Area[tex]=\frac{1}{2} (8+12)\times 7[/tex]
Area[tex]=\frac{1}{2} (20)\times 7[/tex]
Area[tex]=(10)\times 7[/tex]
Area[tex]=70in^2[/tex]
The last choice is correct.
There are 45 members on the board of directors for a certain non-profit institution. if they must elect a chairperson, first vice chairperson, second vice chairperson, and secretary, how many different slates of candidates are possible?
Answer:
1,221,759.
Step-by-step explanation:
This is the number of combinations of 5 from 45.
This = 45! / 40! 5!
= 1,221,759.
Answer:
1,221,759
Step-by-step explanation:
What is 0.36 expressed as a fraction in simplest form? Enter your answer in the box. Repeating 36
Answer: 9/25
Step-by-step explanation: .36 equals 36/100 in decimal form.
Reducing it into its simplest form:
Both numbers are divisible by 4-
36/100 = 9/25 - which is the simplest form.
At school 95% of students arrive on time.If there are 200 students how many are expected to be late
Answer:
10 students (late)Step-by-step explanation:
total students = 200
95% of students arrived on time
5% of students were late.
95% x 200 = 190 students (on time)
5% x 200 = 10 students (late)
-------------------------------------
200 students
The figures below are similar. Compare the larger figure to the smaller. which of the following is the ratio of the areas of the figures?
Answer:
4:3
Step-by-step explanation:
The ratio is 8:6, which simplifies to 4:3.
Answer:
16:9
Step-by-step explanation:
Length of a rectangle is 5 cm longer than the width. Four squares are constructed outside the rectangle such that each of the squares share one side with the rectangle. The total area of the constructed figure is 120 cm2. What is the perimeter of the rectangle?
Answer:
18 cm
Step-by-step explanation:
Let the width of the rectangle be x cm, then the length of the rectangle is x+5 cm.
The area of the ractangle is
[tex]x\cdot (x+5)\ cm^2.[/tex]
The area of the square with side's length of x cm is
[tex]x^2\ cm^2.[/tex]
The area of the square with side's length of x+5 cm is
[tex](x+5)^2\ cm^2.[/tex]
The area of constructed figure is
[tex]x(x+5)+2x^2+2(x+5)^2\ cm^2.[/tex]
Since the total area of the constructed figure is 120 cm², you have
[tex]x(x+5)+2x^2+2(x+5)^2=120.[/tex]
Solve this equation:
[tex]x^2+5x+2x^2+2x^2+20x+50=120,\\ \\5x^2+25x-70=0,\\ \\x^2+5x-14=0,\\ \\D=5^2-4\cdot (-14)=25+56=81,\\ \\x_{1,2}=\dfrac{-5\pm\sqrt{81}}{2}=-7,\ 2.[/tex]
The width of the rectangle cannot be negative, so x=2 cm and x+5=7 cm and the perimeter of the rectangle is
[tex]2x+2(x+5)=2\cdot 2+2\cdot 7=4+14=18\ cm.[/tex]
What are the tangent ratios for Y and Z? The triangle is not drawn to scale. Please help I selected an answer but I’m sure it’s wrong
Answer:
[tex]tanY = \frac{4}{5}[/tex] and [tex]tanZ=\frac{5}{4}[/tex]
Explanation:
The tan of an angle in a triangle is equal to [tex]\frac{opposite}{adjacent}[/tex] for the side lengths. Opposite side is the one found directly across the triangle and the adjacent is the side touching the angle that is not the hypotenuse. This means the tan for Y is 4 over 5 and the tan for Z is 5 over 4.
The tangent ratios are:
tan(Y) = 5/4
tan(Z) = √41/4
In a right triangle with sides labeled X, Y, and Z, and angles labeled x, y, and z, you can use the trigonometric ratios to find the tangent ratios for angles Y and Z.
The tangent ratio is defined as:
For angle Y (opposite to side XY): tan(Y) = opposite / adjacent = XY / ZX
For angle Z (opposite to side ZY): tan(Z) = opposite / adjacent = ZY / XZ
Given the following values:
ZX = 4
XY = 5
ZY = √41
Let's calculate the tangent ratios:
For angle Y (opposite to XY):
tan(Y) = XY / ZX = 5 / 4
For angle Z (opposite to ZY):
tan(Z) = ZY / XZ = √41 / 4
So, the tangent ratios are:
tan(Y) = 5/4
tan(Z) = √41/4
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FIND ST OF THIS CIRCLE.
[tex]\bigtriangleup RSP \cong\bigtriangleup RQT\Rightarrow \frac{RS}{RQ}=\frac{RP}{RT}\Leftrightarrow RS=3 \\ \Rightarrow ST=RT-RS=5[/tex]
The probability assigned to each experimental outcome must be
Answer:
Between 0 and 1
Step-by-step explanation:
The probability for each outcome in experimental probability must be a value between 0 and 1. A probability of 0 means the event did not happen; the probability of 1 means it is certain. We cannot have probabilities smaller than 0 or greater than 1.
All of the probabilities in an experiment must sum to 1 whole.
The probability assigned to each experimental outcome must be greater than 0 or less than 1 or lies in between 0 and 1.
The probability of extremely certain event is 1 and the probability of extremely uncertain event is 0. The sum of all the probabilities in an experiment is always equals to 1.
Hence the probability assigned to each experimental outcome must be greater than 0 or less than 1 or lies in between 0 and 1.
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