The value of f(7.5) of the exponential function is 86.84
How to find the value of f(7.5) of the exponential function?
The general form of an exponential function is given by:
f(x) = abˣ
Since f(x) is an exponential function, and f(4)=21 and f(4.5)=47. We can write that:
21 = ab⁴ --- (1)
47 = ab⁴°⁵ --- (2)
Divide (2) by (1):
47/21 = b⁴°⁵⁻⁴
47/21 = b⁰°⁵
b =√(47/21)
b = 1.50
Put b = 1.5 into (1):
21 = ab⁴
21 = a(1.5)⁴
a = 21/(1.5)⁴
a = 4.15
So the exponential function f(x) = 4.15(1.50)ˣ
To find the value of f(7.5), put x = 7.5 into the exponential function:
f(x) = 4.15(1.50)ˣ
f(7.5) = 4.15(1.50)⁷°⁵
f(7.5) = 86.84
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The functions u and w are defined as follows. u(x)=2x+2 w(x)=-2x^2+2 find the value of w(u(4)).
The value of w(u(4)) is 73.
u( x ) = - 2*x + 2
w( x ) = 2*x^2 + 1
w( u( x ) ) = w( - 2*x + 2 )
⇒ w( u( x ) ) = 2*( - 2*x + 2 )^2 + 1
⇒ w( u( x ) ) = 2*[ 2^2 + ( 2*x )^2 - 2*2*( 2*x ) ] + 1
⇒ w( u( x ) ) = 2*[ 4 + 4*x^2 - 8*x ] + 1
⇒ w( u( x ) ) = 8 + 8*x^2 - 16*x + 1
⇒ w( u( x ) ) = 8*x^2 - 16*x + 9
⇒ w( u( 4 ) ) = 8*4^2 - 16*4 + 9
⇒ w( u( 4 ) ) = 8*16 - 16*4 + 9
⇒ w( u( 4 ) ) = 128 - 64 + 9
⇒ w( u( 4 ) ) = 73
In mathematics, a feature from a set X to a fixed Y assigns to each detail of X exactly one detail of Y. The set X is called the domain of the function and the set Y is referred to as the codomain of the function. features were firstly the idealization of how various quantity relies upon any other quantity.
These elementary functions include
rational functions, exponential functions, basic polynomials, absolute values square root function.A function is defined as a relation between a set of inputs having one output each. In simple phrases, a function is a courting among inputs wherein every entry is associated with exactly one output. every function has a site and codomain or range. A characteristic is normally denoted through f(x) in which x is the input.
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Write the slope-intercept form of the equation of each line
1. 9x -7y = -7
2. 11x -4y = 32
3. 11x- 8y = -48
The slope intercept equation can be represented as follows:
y = 9 / 7 x + 1
y = 11 / 4 x - 8
y = 11 / 8 x + 6
How to write slope intercept equation?The slope intercept equation can be represented as follows:
y = mx + b
where
m = slopeb = y-interceptTherefore,
The slope intercept equation can be represented as follows:
9x - 7y = - 7
7y = 9x + 7
y = 9 / 7 x + 1
11x - 4y = 32
4y = 11x - 32
y = 11 / 4 x - 8
11x - 8y = -48
8y = 11x + 48
y = 11 / 8 x + 6
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this is confusing me please helpppp
Answer:
base = 12 cm , altitude = 16 cm
Step-by-step explanation:
the area (A) of a triangle is calculated as
A = [tex]\frac{1}{2}[/tex] ba ( b is the base and a the altitude )
here a = b + 4 , then
[tex]\frac{1}{2}[/tex] b(b + 4) = 96 ( multiply both sides by 2 to clear the fraction )
b(b + 4) = 192
b² + 4b = 192 ( subtract 192 from both sides )
b² + 4b - 192 = 0 ← in standard quadratic form
(b + 16)(b - 12) = 0 ← in factored form
equate each factor to zero and solve for b
b + 16 = 0 ⇒ b = - 16
b - 12 = 0 ⇒ b = 12
however, b > 0 then b = 12
so base = 12 cm and altitude = b + 4 = 12 + 4 = 16 cm
Answer:
base = 12 cm
Altitude = 16 cm
Step-by-step explanation:
Area of the triangle:[tex]\sf \boxed{Area \ of \ triangle = \dfrac{1}{2}*base*height}[/tex]
base = x cm
altitude or height = (x + 4) cm
[tex]\sf \dfrac{1}{2}*x*(x+4) = 96\\\\[/tex]
x * (x + 4) = 96*2
x*x + x*4 = 192
x² + 4x = 192
x² + 4x - 192 = 0
Sum = 4
Product = -192
Factors = (-12) , 16
When we add (-12) & 16, we get 4. When we multiply (-12) & 16, we get (-192).
Rewrite the middle term using the factors.
x² + 16x - 12x - 192 = 0
x(x + 16) -12(x + 16) = 0
(x + 16)(x - 12) = 0
x - 12 = 0 or x + 16 = 0
x = 12 or x = -16
Ignore x = -16 as measurements won't be in negative value.
x = 12
Base = 12 cm
Altitude = 12 + 4 = 16 cm
I drive from town a to my home and then to town b. the journey time is 50 minutes what is the average speed?
The average speed can be shown as (A + B)/50 miles per minute, where the distance between town A and his home is A miles, and that from his home to town b is B miles, and the total time is 50 minutes.
The average speed of any object is given as the ratio of the total distance traveled by the body, to the total time taken.
Average Speed = Total Distance/Total Time.
In the question, we are informed that the person drove from town a to his home and then to town b, with the total journey time as 50 minutes.
We are asked to find the average speed of the person.
We assume the distance from town a to his home to be A miles.
We assume the distance from his home to town b to be B miles.
Thus, the total distance traveled = A + B miles.
The total time taken by the person is given to be 50 minutes.
We know that the average speed of any object is given as the ratio of the total distance traveled by the body, to the total time taken.
Thus, the average speed of the person can be shown as:
Average Speed = (A + B)/50 miles per minute.
Thus, the average speed can be shown as (A + B)/50 miles per minute, where the distance between town A and his home is A miles, and that from his home to town b is B miles, and the total time is 50 minutes.
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(05.02 MC)
Solve 2 log x = log 64.
Ox= 1.8
Ox=8
Ox= 32
Ox = 128
Answer:
x = 8
Step-by-step explanation:
[tex]2logx=log64\\logx^2=log64[/tex]
now that both sides has log, you can remove them and leave it as:
[tex]x^{2} =64\\[/tex]
take the square root of both sides:
[tex]x = [8, -8]\\x = 8[/tex]
logarithmic equations can't have negative solutions, plus -8 isn't an option
i hope this helped!
Your ship has steamed 1651 miles at 20 knots using 580 tons of fuel oil. The distance remaining to your next port is 1790 miles. If you increase speed to 25 knots, how much fuel will be used to reach that port
The 680 tons of fuel needed to cover the next 1790 miles distance.
According to the statement
We have given that the ship covered a 1651 miles distance at 20 knots by a 580 tons of fuel oil.
And distance remaining to next port is 1790 miles and speed increases to 25 knots.
And we have to find that how much fuel used to reach that port.
So, Distance covered = 1651 mile
fuel used = 580 tons
distance covered with 1 tons fuel = 1651/580
distance covered with 1 tons fuel = 2.85 mile at a speed of 20 knots.
so, The fuel consumption for 1790 mile is = 1790/2.85
The fuel consumption for 1790 mile is = 628 tons of fuel needed.
So, The 680 tons of fuel needed to cover the next 1790 miles distance.
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A triangle is given by A ( 3,2 ) B (-1,5) and C (0,-1) . What is the equation of the line through B parallel to AC ( give ans and how it works pls )
Answer:
Step-by-step explanation:
Solution :
hello :
note :
Use the point-slope formula.
y - y_1 = m(x - x_1) when : x_1= -1 y_1= 5
m= the slope is : (YC - YA)/(XC -XA)
(-1-2)/(0-3) = -3/-3 =1
( parallel to AC means same slope)
an equation in the point-slope form is : y -5 = 1(x+1)
y=x+6
Jenny draws the path to walk from her house over to her grandmother's house. She knows that she has to avoid the end of the segment where construction is taking place. If she wants to use equations to identify her path on the coordinate plane, what should her answer look like?
Her answer should look like y = x + 2, x ≤ 2
How to determine the equation?The complete question is added as an attachment
From the attached graph, we have the following points
(x, y) = (2, 4) and (0, 2)
See that the difference between the y and x values is 2 where y > x
So, we have
y - x = 2
Add x to both sides
y = x + 2
The highest value of x in the graph is 2.
So, we have
x ≤ 2
Hence, her answer should look like y = x + 2, x ≤ 2
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Use substitution to find the integral. (Remember to use absolute values where appropriate.) square root x / x-4
Answer:
Step-by-step explanation:
here is a solution and verify
What is the standard equation for the circle with center (1,-3) that passes through the point (2,2)?
The standard equation for the circle is (x - 1 ) ^2 + (y + 3)^2 = 2 ^2
A circle is a closed curve that is drawn from a fixed point called the center in which all points on the curve are the same distance from the center of the center. The equation of a circle with center (h, k) and radius r is given by:
(x-h)^2 + (y-k)^2 = r^2
This is the standard form of the equation. So if we know the coordinates of the center of the circle and also its radius, we can easily find its equation.
Consider any point P(x, y) on the circle. Let "a" be the radius of the circle which is equal to OP.
We know that the distance between the point (x, y) and the origin (0,0) can be found using the distance formula which is equal to -
√[x^2+y^2]= a
Therefore the equation of a circle with center as origin is,
x^2+y^2= a^2
Where "a" is the radius of the circle.
An alternative method
Let's derive another way. Assume that (x,y) is a point on the circle and the center of the circle is at the origin (0,0). Now, if we draw a perpendicular from the point (x,y) to the x-axis, we get a right triangle, where the radius of the circle is the hypotenuse. The base of the triangle is the distance along the x-axis and the height is the distance along the y-axis. Applying the Pythagorean theorem here, we therefore obtain:
x^2+y^2 = radius^2
We need to find the standard equation for the circle with center (1,-3) that passes through the point (2,2)
Standard equation for circle is
(x-h)^2 + (y-k)^2 = r^2................(1)
Here h = 1 , k = -3 and r = 2
put this value of h, k and r in equation (1), we get
(x - 1 ) ^2 + (y + 3)^2 = 2^2
Hence the standard equation of the circle is
(x - 1 ) ^2 + (y + 3)^2 = 2 ^2
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A rectangular piece of land measures 4.5 km long and 2.5 km wide
Calculate the ratio of
1.It's width to its length
2.Its length to its perimeter
The ratio of its width to its length is; 5 :9.
The ratio of its length to its perimeter is; 9: 28.
What is the ratio of its width to its length?1). From the task content, the land measures 4.5 km long and 2.5 km wide. Hence, the ratios is; 2.5: 4.5.
Therefore, we have; 5: 9 by expression as whole number ratios.
2) Since the perimeter of the rectangle is; 2(2.5+4.5) = 14.
The ratio of length to perimeter expressed simply is therefore; 9: 28.
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Which is the best estimate of -14 1/9 (-2 9/10) ?
Answer:
42
Step-by-step explanation:
-14 1/9 is about -14.
-2 9/10 is about -3.
Multiplying these, we get (-14)(-3)=42.
Which lines can you conclude are parallel given that M<7 m<11=180 Justify your conclusion with a theorem. Line c is parallel to line d by the Converse of the Alternate Interior Angles Theorem. Line c is parallel to line d by the Converse of the Same-Side Interior Angles Theorem. Line a is parallel to line b by the Converse of the Alternate Interior Angles Theorem. Line a is parallel to line b by the Converse of the Same-Side Interior Angles Theorem. 23
Lines a and b are parallel based on the Converse of Same-Side Interior Angles Theorem,
What is the Converse Same-Side Interior Angles Theorem?The Converse of Same-Side Interior Angles Theorem states that, if the sum of two interior angles on same side of a transversal equals 180 degrees, then the lines they lie on are parallel to each other.
Since M<7 + m<11 = 180, lines a and b are parallel based on the Converse of Same-Side Interior Angles Theorem,
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Rearrange the equation so a is the independent variable.
−3a+6b=a+4b
Answer:
6b-4b=a+3a
2b=4a
4a=2b
a=2b/4
a=b/2
John is looking at the top of a building and thinking it would be great to make a zipline from the top to your current location. He measures the angle of elevation as 28and is 182 feet from the base of the buildingHow long a zipline is needed if John is 71.5 inches tall? Include a sketch that shows all known information and clearly shows what you need to findShow all work and give the answer rounded to the nearest tenth of a foot
Answer:
Step-by-step explanation:
sin(28) = 2184 / x
x= 2184/ sin(28) = 2184 / 0.4694 = 4652.75
4652.75 / 12 = 387.7 ft
ANSWER: 387.7ft
In the given case, The length of the zipline needed is approximately 96.7 feet.
To find the length of the zipline needed, we can use trigonometry. Let's start by drawing a sketch to visualize the problem. | | zipline | | | | | | | | | | John | Building 28°
John's location, with the angle of elevation labeled as 28°. John is 182 feet away from the base of the building, and he is 71.5 inches tall.
To find the length of the zipline, we can use the tangent function.
The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
In this case, the opposite side is John's height, and the adjacent side is the distance between John and the building. Using the tangent function, we have: tan(28°) = opposite / adjacent
We want to find the length of the zipline, which is the hypotenuse of the right triangle formed by John, the building, and the zipline. Let's call this length "x".
Therefore, we can rewrite the equation as: tan(28°) = 71.5 inches / 182 feet To solve for x, we need to convert the units so that they match.
Let's convert inches to feet: 71.5 inches = 71.5 inches * (1 foot / 12 inches) ≈ 5.96 feet (rounded to the nearest hundredth)
Now, let's solve for x: tan(28°) = 5.96 feet / 182 feet
To isolate x, we can multiply both sides by 182 feet: x ≈ tan(28°) * 182 feet Using a calculator, we find that tan(28°) ≈ 0.5317.
Multiplying this by 182 feet, we get: x ≈ 0.5317 * 182 feet ≈ 96.7 feet (rounded to the nearest tenth) .
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A survey of several 10 to 11 year olds recorded the following amounts spent on a trip to the mall: $18.31,$25.09,$26.96,$26.54,$21.84,$21.46 Construct the 98% confidence interval for the average amount spent by 10 to 11 year olds on a trip to the mall. Assume the population is approximately normal. Step 3 of 4 : Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.
The 98% confidence interval for the average amount spent by 10 to 11-year-olds on a trip to the mall is 23.36 ± 2.933.
How to calculate the confidence interval and its critical value?The confidence interval for a given level of percentage is given by
C. I = μ ± Z(σ/√n)
Where,
μ - mean, σ - standard deviation, n - sample size, and z - critical value.
The critical value is calculated by
step 1: 100% - (the confidence level)
step 2: Converting the step 1 result into decimal value
step 3: dividing the step 2 result by 2
This is indicated by α/2. So, from the normal distribution table, the z-value at α/2 is said to be the required critical value and denoted by Z_(α/2) or Z.
Calculation:It is given that,
A survey of several 10 to 11-year-olds recorded the following amounts spent on a trip to the mall: $18.31,$25.09,$26.96,$26.54,$21.84,$21.46
Sample size n = 6
Step 1: Finding the mean for the given amounts of the survey:
Mean μ = (18.31 + 25.09 + 26.96 + 26.54 + 21.84 + 21.46)/6
= 23.36
Step 2: Finding the standard deviation:
Standard deviation σ = √summation(x - mean)²/n
On calculating, we get σ = 3.09
Step 3: Finding the critical value:
It is given that the confidence level is 98%
So, (100% - 98%) = 2%
Converting into decimal gives 0.02
So, α/2 = 0.02/2 = 0.01
Thus, at 0.01, the critical value Z = 2.326
Step 4: Constructing the confidence interval:
C.I = 23.36 ± (2.326) × (3.09/√6)
= 23.36 ± (2.326 × 1.261)
= 23.36 ± 2.933
So, the lower bound = 23.36 - 2.933 = 20.427
the upper bound = 23.36 + 2.933 =26.293
Therefore, the 98% confidence interval lies from 20.427 to 26.293.
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Instructions: Find the measure of the indicated angle to the nearest degree.
Pls help me !!! due today:(
Answer:
x = 20 units
Step-by-step explanation:
See attached.
Answer:
[tex]x = 94[/tex]
Step-by-step explanation:
To find the value of [tex]x[/tex], we have to first find the lengths of the sides of the squares and add them.
The formula for area of a square is as follows:
[tex]{area = (length \space\ of \space\ side)^2}[/tex]
This means the length of a side of a square can be found by:
[tex]\boxed {length \space\ of \space\ side = \sqrt{area}}[/tex]
Now we can find the lengths of sides of each of the squares:
• Length of side of larger square = [tex]\sqrt{8100}[/tex]
= 90
• Length of side of smaller square = [tex]\sqrt{16}[/tex]
= 4
Next we can just add the lengths of the sides to get the value of [tex]x[/tex] :
[tex]x = 90 + 4[/tex]
⇒ [tex]x \bf = 94[/tex]
each of 16 students in a class made a poetry book. each book contained 24 poems. how many poems are in 16 books
HELP MEEEEEEEEEEEEEEEE
Step-by-step explanation:
Well first let's find the derivative in using the power rule
h'(t) = (-4.9)t + 157
h'(t) = -9.8t + 157
we can see from the image above that the maximum height is 1452.602 so that is the answer to the second question.
h'(1452.602) = -9.8(1452.602) + 157
h' = 14,078.500
Identify each expression that represents the slope of a tangent to the curve y = 1/x+1 at any point (x,y)
The expressions that represent the slope of a tangent to the curve [tex]y = \frac{1}{x+1}[/tex] are [tex]m = \lim_{h \to 0}\frac{\frac{1}{x + h + 1} - \frac{1}{x + 1} }{h}[/tex] and [tex]m = -\frac{1}{(x + 1)^{2}}[/tex], respectively.
How to find an expression for the slope at any point of a function
The slope at any point of the function can be found by definition of derivative following algebraic handling:
[tex]m = \lim_{h \to 0}\frac{\frac{1}{x + h + 1} - \frac{1}{x + 1} }{h}[/tex]
[tex]m = \lim_{h \to 0} \frac{\frac{x+1 - x - h - 1}{(x + h + 1)\cdot (x + 1)} }{h}[/tex]
[tex]m = -\lim_{h\to 0} \frac{1}{(x + h + 1)\cdot (x + 1)}[/tex]
[tex]m = -\frac{1}{(x + 1)^{2}}[/tex]
Thus, the expressions that represent the slope of a tangent to the curve [tex]y = \frac{1}{x+1}[/tex] are [tex]m = \lim_{h \to 0}\frac{\frac{1}{x + h + 1} - \frac{1}{x + 1} }{h}[/tex] and [tex]m = -\frac{1}{(x + 1)^{2}}[/tex], respectively.
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HELP PLS 5. Find the missing side lengths. Leave the answers as radicals in simplest form.
Answer:
[tex]a=22, b=11\sqrt{3}[/tex]
Step-by-step explanation:
Given a 30-60-90 triangle, the shortest side is always opposite of the smallest angle, in this case 11 is opposite the 30° angle. The largest side of a 30-60-90 triangle will always be twice the smallest, giving [tex]a[/tex] a measure of [tex]22[/tex]. To get the side across the 60° angle we multiply the smallest side by the square root of 3. Meaning [tex]b=11\sqrt{3}[/tex]
In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and the other 200,745 children were given a placebo. Among those in the treatment group, 33 developed polio, and among those in the placebo group, 115 developed polio.
The correct option regarding whether the requirements for a hypothesis test are satisfied is:
D. The conditions are satisfied. The samples are random, and each sample has at least 5 successes and 5 failures.
What are the conditions for an hypothesis test?There are two conditions:
The samples should be random.Each sample should have at least 5 successes and at least 5 failures.These two conditions are satisfied for this problem, hence, researching this problem on the internet, option D is correct.
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Write an equation of the line that passes through the point (8, 1) with slope 5. A. y - 1 = 5(x − 8) B. y − 1 = - -5(x − 8) - C. y - 8 = -5(x - 1) OD. y - 8 = 5(x − 1) Question Progress
Answer:
[tex]y-1=5(x-8)[/tex]
Choose the correct simplification of the expression ( 4x/y)^2.
Answer options can be found in photo!
[tex]\large\displaystyle\text{$\begin{gathered}\sf \left(\frac{4x}{y}\right)^{2} \end{gathered}$}[/tex]
To raise 4x/y to a power, raise the numerator and denominator to the power, and then divide.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \frac{(4x)^{2} }{y^{2} } \end{gathered}$}[/tex]
Expand (4x)².
[tex]\large\displaystyle\text{$\begin{gathered}\sf \frac{4^{2} x^{2} }{y^{2} } \end{gathered}$}[/tex]
Calculates 4 to the power of 2 and gets 16.
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \frac{16x^{2} }{y^{2} } \end{gathered}$} }[/tex]
a bar of dark chocolate is advertised to contain 60 % of pure cocoa the total weight of the bar is 8 ounces how many ounces if the bar are pure cocoa
Answer:
4.8
Step-by-step explanation:
60 percent of 8 is 4.8
I hope this helps!
What is the height of a cylinder with a volume of 315 pi cubic meters and a radius of 6 meters?
8.75 meters
12.25 meters
26.25 meters
52.5 meters
The height of the cylinder is (a) 8.75 meters
How to determine the height?The given parameters are:
Volume = 315[tex]\pi[/tex]
Radius, r = 6
The volume of a cylinder is:
[tex]V = \pi r^2 h[/tex]
So, we have:
[tex]315\pi = \pi * 6^2 * h[/tex]
Divide both sides by 36[tex]\pi[/tex]
h = 8.75
Hence, the height of the cylinder is (a) 8.75 meters
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A cylinder has a volume of 200 mm³ and a height of 17 mm.
a) The volume formula for a cylinder is V = r²h. Isolate for the variable r in this formula.
b) Using the equation where you isolated for r in part a, find the radius of the cylinder.
Round your answer to the nearest hundredth.
The radius of the cylinder exists 1.93mm.
How to estimate the radius of the cylinder?Let V be the volume of the cylinder exists 200 mm³
r be the radius
h be the height exists 17
Volume of cylinder, V = π r²h
200 = π r² (17)
200 = 53.40707 r²
200 = 53.41 r²
simplifying the above equation, we get
r² = 200/53.41
r² = 3.74461 = 3.74
r² = 3.74
r = 1.933907961 = 1.93
Therefore, r = 19.3
So the radius of the cylinder given the volume exists 200 mm³ and a height of 17 mm exists 1.93 mm.
Therefore, the radius of the cylinder exists 1.93mm.
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so confused pls help>>>> there’s a pic
Answer:
(c) 25/4
Step-by-step explanation:
A perfect square trinomial is obtained by "completing the square." The constant in a perfect square trinomial is the square of half the x-coefficient.
Square of a binomialA "perfect square trinomial" is the square of a binomial.
(x +a)² = x² +2ax +a²
Of note here is that the constant (a²) is the square of half of the x-term coefficient:
((2a)/2)² = a²
Completing the squareThe given expression (x² +5x) has an x-term coefficient of 5. The square of half that is ...
(5/2)² = 25/4
The constant that must be added to make x² +5x into a perfect square trinomial is 25/4.
(x² +5x) +25/4 = x² +5x +25/4 = (x +5/2)²
Select the correct answer.
(1, √3) to polar form.
A. (√3, 30°)
B. (2,60°)
C. (4,30°)
D. (2,240°)
Answer:
(2, 60°)
Step-by-step explanation:
A point in polar form (a,b) is represented by a radius and an angle (r, ϴ). To convert, use the formula r² = a² + b² to get r, and [tex]tan^{-1}[/tex](b/a) = ϴ
[tex]r = \sqrt{1^{2} + \sqrt{3} ^{2} }[/tex]
[tex]r = \sqrt{1+3}[/tex]
[tex]r = \sqrt{4}[/tex]
= 2
[tex]tan^{-1}[/tex]([tex]\sqrt{3}[/tex]/1) = ϴ
= 60°