If f(x) = 5x+27, find the instantaneous rate of change of f(x) at x=9
The instantaneous rate of change refers to the derivative of f(x), so let's find that first, following the power rule, we have.
[tex]f^{\prime}(x)=1\cdot5x^{1-1}=5x^0=5\cdot1=5[/tex]As you can observe, the instantaneous rate of change is a constant, which means we don't have to evaluate it at x=9.
Therefore, the instantaneous rate of change is f'(x) = 5.On a coordinate plane, each point on ST is defined by coordinates (x,y). The segmentis rotated 90° counterclockwise to create S'T'. Which of the following defines a pointon S'T' that corresponds to a point on ST.
The coordinates of S'T' are (-y,x)
Here, we want to get the result of rotating a point 90 degrees counterclockwisely
What we have here is that we will have a change in the coordinates of the point as follows;
[tex](x,y)\Rightarrow\text{ (-y,x)}[/tex]The Jurassic Zoo charges $5 for each adult admission and $3 for each child. The total bill for the 243 people from a school trip was $823. How many adults and how many children went to the zoo? There were adults and children on the trip
Let x be the number of adult and let y be the number of children
5x + 3y = 823 --------------------------(1)
x + y = 243 ----------------------------(2)
We can now solve simultaneously
Multiply through equation (2) by 3
3x + 3y =729 -----------------(3)
subtract equation (3) from equation (1)
2x =94
Divide both-side of the equation by 2
x =47
substitute equationx = 47 into equation (2) and then solve for y
47 + y = 243
subtract 47 from both-side of the equation
y = 243 - 47
y = 196
Hence
Which statements are true for the function y= \xi - 2? Select all that apply. The value of the function is never negative. Its graph has a V-shape There is only one input for which the output is 0. There are two inputs for which the output is 5 The vertex. of ts graph 13 81 (0.-2).
Explanation:
We need to check out each of the options to determine which of the statements are true:
The given function: y = |x| - 2
a) The symbol '| |' means absolute. So any number you put in it either positive or negative number,becomes positive after it is removed.
For example: if x = -2, y = |-2| - 2 = 2 -2 = 0
if x = -1, y = |-1| -2 = 1 -2 = -1
Hence from the example above, the value of the function can be negative.
Statement is wrong
b) Yes, the graph has a V shape: (statement is correct)
c) For the output to be 0, it means y = 0
Let's find out the value of x when y= 0
y = |x| -2
0 = |x| -2
0 + 2 = |x|
|x| = 2
|x| = x or -x
From the above we have only one x value (the input) which is 2
Statement is correct
d) For the output to be 5, y = 5
5 = |x| -2
|x| = -2 -5
|x| = -7
|x| = x or -x
x = -7 or -x = -7
x = -7 or x = 7
eleven is fifteen more than four times a number.
translate to an equation. let x be the unknown number . solve for x
The equation is 15+4x= 11 and the value for x= -1
what are equations?An equation is a mathematical statement between two expressions that have equal values. For example, 15+4x= 11. The symbol = shows that 11 and 15+4x are the same
Word problems can be translated into equation with variables.
Representing the unknown with x,
4×x= 4x
the word problem is therefore translated to
11= 15+4x
collecting like terms
4x= 11-15
4x= -4
divide both side by 4
x=-1
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Three. 5 m away from the base of the building. How I is window B?
Explanation
Given that each ladder is 3.5 meters away from the base of the building, we are asked to find the height of the building. This can be seen below,
Number 1:
The height of ladder A is 5.5 meters. We can then use the Pythagoras theorem to find the height of the window.
[tex]\begin{gathered} 5.5^2=3.5^2+A^2 \\ A^2=5.5^2-3.5^2 \\ A=\sqrt{5.5^2-3.5^2}=\sqrt{30.25-12.25} \\ A=4.24m \end{gathered}[/tex]
Answer: Height of window A =4.24m
Number 2
The height of ladder B is 8.8 meters. We can then use the Pythagoras theorem to find the height of the window.
[tex]\begin{gathered} 8.8^2=3.5^2+B^2 \\ B^2=8.8^2-3.5^2 \\ B=\sqrt{8.8^2-3.5^2}=\sqrt{77.44-12.25}=\sqrt{65.19} \\ B=8.07 \end{gathered}[/tex]Answer: Height of window B =8.07m
Number 3
The height of ladder B is 9.3 meters. We can then use the Pythagoras theorem to find the height of the window.
[tex]\begin{gathered} 9.3^2=3.5^2+C^2 \\ C^2=9.3^2-3.5^2 \\ C=\sqrt{9.3^2-3.5^2}=\sqrt{86.49-12.25}=\sqrt{74.24} \\ C=8.62 \end{gathered}[/tex]Answer: Height of window C =8.62m
Number 4
The height of ladder B is 7.9 meters. We can then use the Pythagoras theorem to find the height of the window.
[tex]\begin{gathered} 7.9^2=3.5^2+D^2 \\ D^2=7.9^2-3.5^2 \\ D=\sqrt{7.9^2-3.5^2}=\sqrt{62.41-12.25}=\sqrt{50.16} \\ D=7.08 \end{gathered}[/tex]Answer: Height of window D = 7.08m
Which is an x-intercept of the graphed function?
O (0, 4)
(-1, 0)
(4, 0)
(0, -1)
A line's x-intercept and y-intercept are the points at which the x axes- and y-axes, respectively, are crossed. We find it easier to graph linear equations when we consider intercepts.
We set y = 0 and solve the equation for x to determine the x-intercept.
The graphed function derived from the choices has the x-intercept (-1, 0)
How do you calculate the x-intercept?A graph's x-intercept is the location where the graph crosses the x-axis.
The graph in the accompanying image crosses the x-axis at:
(-2, 0), (-1, 0), (1, 0) and (2, 0) (2, 0)
Consequently, the x-intercept of the graphed function derived from the alternatives (-1, 0)
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Im stuck on question 5-91 in cpm geometry the question asks me to find the area of a triangle... I figured out the 30 60 90's are but i still have triangle left over ill send a photo of the triangle
Answer
Area of the triangle = 72 square units
Explanation
For a triangle whose two sides (a and b) are given, and an angle that exists between them is also given, the area of that triangle is given as
Area of the triangle = ½ ab Sin θ
For this question,
a = 12 units
b = 24 units
θ = 30°
Area of the triangle = ½ ab Sin θ
Area of the triangle = ½ (12) (24) (Sin 30°)
Sin 30° = 0.5
Area of the triangle = ½ (12) (24) (Sin 30°)
Area of the triangle = ½ (12) (24) (0.5) = 72 square units
Hope this Helps!!!
Due to temporary tax cuts in 2010, a person with typical deductions earning $50,000 per year would have saved 2% of their income plus $850 in federal taxes. How much money did a typical person save?
If the person earns $50,000, we must find how much is 2% of that and then add it to $850. Then we would have how much a typical person saves.
First let's find 2% of $50,000
[tex]\begin{gathered} 2\%\text{ of }50000=\frac{2}{100}\cdot500000=\frac{2\cdot50000}{100}=\frac{100000}{100}=1000 \\ \\ \\ \end{gathered}[/tex]Therefore 2% of $50,000 is $1000. Now we add it to $850 and we get
[tex]1000+850=1850[/tex]The final result is $1850
Complete the description of how to find the sums −4 + 1 and −4 + (−1) on a number line.
To find −4 + 1, start at −4 and move 1 unit to the _____ to _____.
To find the sum −4 + (−1), start at −4 and move 1 unit to the ______ to ______
Integer Sums on a Number Line,
Start in the location that the first number denotes. Move the equivalent amount of units to the left if the second number is negative. Move that many units if it's affirmative.
How can I calculate the total on a number line?
Result for an image Fill in the blanks to describe how to locate the sums 4 + 1 and 4 + (1) on a number line. Start at 4, then move one unit to the ____ to ____ to find 4 + 1. Start at 4, then move 1 unit to the _____ to _____ to find the sum 4 + (1).
Utilizing a number line, combine two integers:
Make a number line first.
Next, locate the first integer's position on the number line.
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Jonathan needs to buy some food for his dog. At the nearest store, 6 bags of dog food cost $30.72. How much would Jonathan spend on 4 bags?
Group of answer choices
$15.36
$20.48
$25.60
$22.40
Answer: B
Step-by-step explanation: find the amount for one bag, by doing 30.72/6, to get 5.12. Then multiply that by 4 to get $20.48
Answer: $20.48
Step-by-step explanation:
first divide the total ( 30.72) by six and you will get 5.12 which is how much it cost to by one bag now since they want to know how much it would cost if you bought four bags multiply that by four and you will get 20.48 which is the correct answer
Please help it’s due tonight I will give a brainlist
The percentage of the markdown amount is approximately 30%.
What is the percentage?A percentage is a number or ratio expressed as a fraction of 100 in mathematics. Although the abbreviations "pct.", "pct.", and occasionally "pc" is also used, the percent sign, "%," is frequently used to indicate it. A percentage is a number without dimensions and without a standard measurement. By dividing the value by the total value and multiplying the result by 100, one can determine the percentage. The percentage calculation formula is (value/total value)100%.So, the percentage of markdown amount:
The total amount is $54.88.The markdown amount is $16.38.So, the markdown percentage is calculated as follows:
16.38/54.88 × 1000.2984 × 10029.84Rounding off: 30%
Therefore, the percentage of the markdown amount is approximately 30%.
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Find the value of a.
The value of a is √10/2 if |x| = 2√2 and a|x| = 2√5 in the given Functions respectively.
What is function?Functions are the foundation of calculus in mathematics. The functions are the various types of relationships. In mathematics, a function is represented as a rule that produces a unique output for each input x.
In mathematics, mapping or transformation is used to denote a function. These functions are usually represented by letters like f, g, and h. The domain is defined as the set of all possible values for which the function can be defined.
We have given that f(x) = |x|
Where x is hypotenuse the triangle made by the line
The formula for hypotenuse = c² = a² + b²
Here each side = 2 units
so hypotenuse is
c² = 4² + 4²
c² = 16 + 16
c² = 32
c = 4√2
Thus, f(x) = 2√2
In f(x) = a|x|
The blue lines with the base and side also makes a right angles triangle and the blue line is the hypotenuse
c² = 4² + 2²
c² = 16 + 4
c² = 20
c = 2√5
Thus, f(x) = 2√5
Now, if |x| = 2√2 and a|x| = 2√5
Taking |x| in common we get the equation
2√2 = (2√5)/a
a = (2√5)/(2√2)
a = √5/√2
a = √10/2
Thus, the value of a is √10/2 if |x| = 2√2 and a|x| = 2√5 in the given Functions respectively.
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Write an inequality for the following problem. Use M for your variable.
22 less than 5 times a number is greater than 65.
Helppp
Answer:
5m - 22 > 65
I hope this helps!
A rectangular backyard has a length of of 68 feet and a width if 40 1/2 feet. the owner wants to play 75% of the yard with grass seed the planting rate for the seed is 1.5 pounds per 1000 square feet of area.To the nearest tenth of a pound ,how much grass seed will tye owner need to plant
We have the following:
The first thing to do is calculate the area of the backyard
[tex]\begin{gathered} A=l\cdot w \\ l=68 \\ w=40\frac{1}{2} \\ \text{replacing:} \\ A=68\cdot40\frac{1}{2}=2754 \end{gathered}[/tex]Now, they tell us it is 75% of the backyard therefore
[tex]2754\cdot0.75=2065.5[/tex]Now we calculate the amount in pounds knowing that for each square foot of area 1.5 pounds are needed
[tex]2065.5\cdot\frac{1.5}{1000}=3.1[/tex]The answer is 3.1 pounds
The mean of a set of data is 25 with a standard deviation of 2. What is the interval about the mean of the data within one standard deviation? Where is my answer?
Mean = 25
Standard deviation= 2
The interval about the mean is
Thus the interval will be
(mean - SD) to (mean + SD)
(25-2) to ( 25 +2)
23 to 27
The interval will be between 23 - 27
change the height for Ramp 2 to make a smooth slide
First answer:
If you look carefully to the measures of the three triangles that are the ramps of the exercise, you will notice they are similar after comparing them.
• Ratio of Ramp 1 = 2/3
,• Ratio of Ramp 2 = 4/6
,• Ratio of Ramp 3 = 6/9
Second answer
The information is given on the image and we can calculate the missing measures for make a smooth slide:
• Ramp 1 - Base 10 - Height 12
,• Ramp 2 - Base 15 - Height 18
,• Ramp 3 - Base 30 - Height 36
We used the ratio 2/3 for answering the first question, now we have the ratio
5/6, as follows:
• Ramp 1 - 5/6 * 2/2 = 10/12
,• Ramp 2 - 5/6 * 3/3 = 15/18
,• Ramp 3 - 5/6 * 6/6 = 30/36
What is 1/4 x 3
Give it's answer in its simplest form
================================================
Explanation:
Think of 3 as 3/1. Then multiply the numerators together separate from the denominators being multiplied together
(1/4)*(3/1) = (1*3)/(4*1) = 3/4
------------------
An alternative route
1/4 x 3 means we have 3 copies of 1/4 being added
1/4 + 1/4 + 1/4 = (1+1+1)/4 = 3/4
------------------
If you are a visual learner, draw a circle and split it into 4 equal pizza slices. Shade 1 out of the 4 to represent 1/4.
Imagine that you did this 3 times. That would mean you have 3 shaded slices. Each slice represents 1/4, so you would have 3/4 of the total pizza.
Put another way: you have 3 friends who get 1/4 of a pizza each. That shades 3 out of the 4 total slices to get 3/4.
Find the intersection points when the line y = 12-4x meets the curve y =12-x^3. Hence, sketch in the same diagram, the line y = 12 - 4x and the curve y=12-x^3 Calculate the area of the region bounded by the curve and the line
The equations of the line and the curve are
[tex]\begin{gathered} y=12-4x \\ y=12-x^3 \end{gathered}[/tex]We will equate their right sides to find the values of x
[tex]12-4x=12-x^3[/tex]Subtract 12 from both sides
[tex]\begin{gathered} 12-12-4x=12-12-x^3 \\ -4x=-x^3 \end{gathered}[/tex]Divide both sides by -1
[tex]\begin{gathered} \frac{-4x}{-1}=\frac{-x^3}{-1} \\ 4x=x^3 \end{gathered}[/tex]Subtract 4x from both sides
[tex]\begin{gathered} 4x-4x=x^3-4x \\ 0=x^3-4x \end{gathered}[/tex]Switch the two sides
[tex]x^3-4x=0[/tex]Take x as a common factor
[tex]\begin{gathered} x(\frac{x^3}{x}-\frac{4x}{x})=0 \\ x(x^2-4)=0 \end{gathered}[/tex]Factor the bracket using the rule of the difference between two squares
[tex](a^2-b^2)=(a+b)(a-b)[/tex][tex]\begin{gathered} (x^2-4)=(x+2)(x-2) \\ x(x+2)(x-2)=0 \end{gathered}[/tex]Now, equate each factor by 0
[tex]\begin{gathered} x=0 \\ x+2=0,x-2=0 \\ x=-2,x=2 \end{gathered}[/tex]Substitute the values of x in the equation of the line to find their corresponding values of y
[tex]\begin{gathered} y=12-4(0)=12 \\ y=12-4(-2)=12+8=20 \\ y=12-4(2)=12-8=4 \end{gathered}[/tex]The points of intersection between the line and the curve are
(0, 12), (-2, 20), (2, 4)
Let us draw the graph
The red line represents the equation of the line
The blue curve represents the equation of the curve
To find the area bounded between the line and the curve we will use the integration
Since the line is above the curve at the point (-2, 20), then
We will subtract the curve from the line and use the values of x -2 and 2 as the limits of integration
[tex]A=\int_{-2}^2[(12-4x)-(12-x^3)]dx[/tex]Simplify the terms inside the brackets first.
[tex]\begin{gathered} A=\int_{-2}^2[12-4x+12+x^3]dx \\ A=\int_{-2}^2[-4x+x^3]dx \end{gathered}[/tex]In integration, we will add the power by 1 and divide the term by the new power
[tex]A=[\frac{-4x^{1+1}}{1+1}+\frac{x^{3+1}}{3+1}]_{-2}^2[/tex]Simplify it
[tex]\begin{gathered} A=[\frac{-4x^2}{2}+\frac{x^4}{4}]_{-2}^2 \\ A=[-2x^2+\frac{x^4}{4}]_{-2}^2 \end{gathered}[/tex]Substitute x by 2 and -2, then subtract the answer
[tex]A=[-2(2)^2+\frac{2^4}{4}]-[-2(-2)^2+\frac{(-2)^4}{4}][/tex]Solve each bracket
[tex]A=8[/tex]The area of the region bounded between the line and the curve is 8 square unit
Express the statement in symbolic form and construct the truth table for the statement. "Cheap Trick does not play at the White House and Jacque is the president."Use the symbolic representationsp: cheap trick plays at the White Houseq: Jacque is the president(May complete on your paper or in the space provided)
Answer:
Explanation:
Here, we want to express the statement in symbolic form and construct the truth table for the statement
From what we have, the symbolic form is as follows:
[tex]\text{ \textasciitilde p }\wedge\text{ q}[/tex]Now, let us complete the truth table as follows:
the measure of an interior angle of a regular polygon is given. find the number of sides in a polygon. show work. Number 8.
Given the following interior angle of a regular polygon:
[tex]120\degree[/tex]You need to remember that, by definition, a regular polygon is a polygon whose sides have all equal lengths.
Therefore, you can apply the following formula:
Where "n" is the number of sides of the polygon and β is the measure of one interior angle of the polygon.
Knowing that, in this case:
[tex]\beta=120\degree[/tex]Therefore, you can substitute this value into the formula and solve for "n":
[tex]\begin{gathered} 120=\frac{(n-2)\cdot180}{n} \\ \\ 120n=180n-360 \end{gathered}[/tex][tex]\begin{gathered} 120n-180n=-360 \\ \\ -60n=-360 \\ \\ \\ n=\frac{-360}{-60} \end{gathered}[/tex][tex]n=6[/tex]Hence, the answer is:
[tex]n=6[/tex]
What is the slope of BC in the simplest form?
Explanation:
The slope of BC can be calculated
Slope = rise/run
In this case, the rise is the height of the blue triangle and the run is the base of the blue triangle, so
Slope = 2/6 = -2/6
To simplify the slope, we need to divide by 2, so
[tex]\text{Slope}=-\frac{2\div2}{6\div2}=-\frac{1}{3}[/tex]What is the volume of the pyramid shown below?A. 294 cu. unitsB. 1008 cu. unitsC. 112 cu. unitsD. 196 cu, units
Step 1
Write an expression for the volume of a square-based triangular pyramid
[tex]\text{Volume(V) =}\frac{1}{3}\times base\text{ area(b)}\times height[/tex]height= 12
Step 2
Find the base area
[tex]\begin{gathered} \text{Base area = length }\times\text{ width} \\ \text{Length = 7} \\ \text{Width = 7} \\ \text{Base area = 7}\times7=49unit^2 \end{gathered}[/tex]Step 3
Find the volume of the shape
[tex]\begin{gathered} V\text{ = }\frac{1}{3}\times49\times12 \\ V\text{ =}196unit^3 \end{gathered}[/tex]You are reducing a map of dimensions 24 in. by 36 in. to fit onto a piece of paper 8 in. by 10 in. What are the dimensions of the largest possible map that can fit on the page?
We have
scale factor
[tex]\frac{24}{36}=\frac{2}{3}[/tex]The largest side 10 in
Then we need to obtain the shortest side
[tex]\frac{2}{3}=\frac{x}{10}[/tex][tex]x=\frac{2\cdot10}{3}=\frac{20}{3}=6\frac{2}{3}[/tex]Therefore the dimensions of the largest possible map are
6 2/3 in by 10 in
(x-8)(2x+5)=0 solve the equation
Solution:
Given the equation:
[tex](x-8)(2x+5)=0[/tex]To solve the equation, we use the zero factor principle.
From the zero factor principle, we have
[tex]\begin{gathered} When\text{ } \\ ab=0 \\ \Rightarrow a=0\text{ or b=0} \end{gathered}[/tex]Thus, we have
[tex]\begin{gathered} x-8=0\text{ or 2x+5 =0} \\ when \\ x-8=0 \\ add\text{ 8 to both sides of the equation} \\ x-8+8=0+8 \\ \Rightarrow x=8 \\ when \\ 2x+5=0 \\ add\text{ -5 to both sides of the equation,} \\ 2x+5-5=0-5 \\ \Rightarrow2x=-5 \\ divide\text{ both sides by the coeffient of x, which is 2} \\ \frac{2x}{2}=\frac{-5}{2} \\ \Rightarrow x=-\frac{5}{2} \end{gathered}[/tex]Hence, the solution to the equation is
[tex]x=8,\text{ x=-}\frac{5}{2}[/tex]The circle below is centered at (10,4) and has a radius of 4. What is itsequation?101010O A. (x - 10)2 + (y - 4)2 = 16O B. (x-4)2 + (-10)2 = 4O C. (x-4)2 + (-10)2 = 16OD. (x-10)2 + (x-4)2 = 4
The general equation of a circle with centre (h,k) and radius r is given as
[tex]\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ \text{where,} \\ (h,k)=(10,4) \\ r=4 \end{gathered}[/tex]By substitution, we will have,
[tex]\begin{gathered} (x-10)^2+(y-4)^2=4^2 \\ =(x-10)^2+(y-4)^2=16 \end{gathered}[/tex]Hence,
The correct answer is OPTION A
In parallelogram FGHJ if m^GHJ=55° find m ^ FGH. I
Given:
[tex]m\angle GHJ=55^o\text{ and m}\angle FGH=x^o[/tex]Recall that the adjacent angles of the parallelogram are supplementary.
[tex]\begin{gathered} m\angle GHJ\text{ and m}\angle FGH\text{ are adjacent angles of the given parallelogram and these are supplementary.} \\ \end{gathered}[/tex]We know that the sum of the supplementary angles is 180 degrees.
[tex]m\angle GHJ+\text{m}\angle FGH=180^o[/tex][tex]55^o+x^o=180^o[/tex][tex]x^o=180^o-55^o[/tex][tex]x^o=125^o[/tex]Hence the answer is
[tex]\text{ m}\angle FGH=125^o[/tex]a right triangle ABC is shown below the area of the triangle above will equal 1/2 of the rectangle that is 5 units long and underscore units wide
for the ahe area of triangle
Height of the triangle = 2 units, base= 5 unit
Area of triangle =1/2 Breadth x height
Susbtitute the value and simplify:
[tex]\begin{gathered} \text{ area of triangle=}\frac{1}{2}\times2\times5 \\ \text{ area of triangle= }5\text{ units} \end{gathered}[/tex]Answer =5 units
Find all Polar coordinates of point P where P= (5,-π/6)
Given :
[tex]P\text{ = (5, }\frac{-\pi}{6})[/tex]Required: All polar coordinates of the given point
We can have the following points:
[tex]\begin{gathered} (5\text{ , }\frac{-\pi}{6}\text{ + 2n}\pi)\text{ or (-5 ,( }\pi-\frac{-\pi}{6}+\text{ 2n}\pi)\text{ )} \\ \text{This simplifies to } \\ (5\text{ , }\frac{-\pi}{6}\text{ + 2n}\pi)\text{ or (-5, (}\frac{-\pi}{6}\text{ + (2n + 1)}\pi) \end{gathered}[/tex]The correct option is the third option
A home has a triangular backyard. The second angel of the triangle is 3 more than the first angel. The third angel is 9 more than four times the first angel.
The measure of three angles of triangular backyard are 28°, 31° and 121°.
Given that, a home has a triangular backyard.
What is the angle sum property of a triangle?Angle sum property of triangle states that the sum of interior angles of a triangle is 180°.
Let the measure of first angle be x.
The second angel of the triangle is 3 more than the first angel = x+3
The third angel is 9 more than four times the first angel = 4x+9
Using angle sum property of a triangle
x+x+3+4x+9 = 180°
⇒ 6x + 12 = 180
⇒ 6x = 168
⇒ x = 28°
So, x+3 = 31° and 4x+9 = 121°
Therefore, the measure of three angles of triangular backyard are 28°, 31° and 121°.
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