In the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be D. the the sum of the area of the square and the area of the circle from a cross section.
How to illustrate the information?It should be noted that a volume simply means the amount of three dimensional space that's enclosed by a closed surface.
It is typically meaured on cubic units.
In this case, in the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be the the sum of the area of the square and the area of the circle from a cross section.
In conclusion, the correct option is D.
Learn more about cone on:
brainly.com/question/28184308
#SPJ1
Answer:
B. It is the ratio of the area of the circle to the area of the square from a cross section.
Step-by-step explanation:
Help me with this question please. ASAP!
Answer:
Step-by-step explanation:
I am going to be honest here. I know the answer is 22 but I cant really explain it you kinda just have to trust I'm right.
Allison went to the grocery store and bought bottles of soda and bottles of juice. Each bottle of soda has 45 grams of sugar and each bottle of juice has 20 grams of sugar. Allison purchased a total of 11 bottles of juice and soda which collectively contain 445 grams of sugar. Write a system of equations that could be used to determine the number of bottles of soda purchased and the number of bottles of juice purchased. Define the variables that you use to write the system.
Using a system of equations, it is found that Allison bought 2 bottles of juice and 9 bottles of soda.
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
For this problem, the variables are:
Variable x: bottles of juice purchased.Variable y: bottles of soda purchased.Allison purchased a total of 11 bottles of juice, hence:
x + y = 11 -> x = 11 - y.
These 11 bottles contain 445 grams of sugar, hence, considering the amounts of each bottle, we have that:
20x + 45y = 445
Since x = 11 - y:
20(11 - y) + 45y = 445
25y = 225
y = 225/25
y = 9.
x = 11 - y = 11 - 9 = 2.
She bought 2 bottles of juice and 9 bottles of soda.
More can be learned about a system of equations at https://brainly.com/question/24342899
#SPJ1
How is the Gauss-Jordan elimination method different from the Gaussian elimination method?
The Gauss-Jordan elimination method different from the Gaussian elimination method in that unlike the Gauss-Jordan approach, which reduces the matrix to a diagonal matrix, the Gauss elimination method reduces the matrix to an upper-triangular matrix.
What is the Gauss-Jordan elimination method?
Gauss-Jordan Elimination is a technique that may be used to discover the inverse of any invertible matrix as well as to resolve systems of linear equations.
It is based on the following three basic row operations that one may apply to a matrix: Two of the rows should be switched around. Multiply a nonzero scalar by one of the rows.
Learn more about Gauss-Jordan elimination method:
https://brainly.com/question/12090959
#SPJ1
A data set has a mean of x = 75 and a standard deviation of 11.4. Find the z-score for each of the following.
Part (a)
[tex]z=\frac{85-75}{11.4} \approx \boxed{0.88}[/tex]
By similar logic, the answers to the other parts are
(b) 1.75
(c) -2.19
(d) 0
consider the function y=-2-cos(x-pi). What effect does pi have on the basic graph?
Using translation concepts, it is found that pi is the phase shift of the graph, and since it is negative, the graph is shifted right pi units.
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.
In this problem, the change is given as follows:
x -> x - pi
It means that the change is in the domain, in which pi is the phase shift of the graph, and since it is negative, the graph is shifted right pi units.
More can be learned about translation concepts at https://brainly.com/question/4521517
#SPJ1
Find all the missing values to make the equation true
The missing value in the logarithm are as follows:
log₃ 7 - log₃ 2 = log₃ (7 / 2)log₉ 7 + log₉ 4 = log₉ 28log₆ 1 / 81 = - 4 log₆ 3How to solve logarithm?Using logarithm rule,
logₐ b - logₐ c = logₐ (b / c)
logₐ b + logₐ c = logₐ (b × c)
Therefore,
log₃ 7 - log₃ 2 = log₃ (7 / 2)
log₉ 7 + log₉ 4 = log₉ (7 × 4) = log₉ 28
log₆ 1 / 81 = log₆ 81⁻¹ = log₆ 3⁻⁴ = - 4 log₆ 3
learn more on logarithm here: brainly.com/question/24515212
#SPJ1
Can someone please help me with this question asap!?
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
HJ = 23.5 in[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
Take HJ = a, GH = b and GJ = c
a = b + 2c = a + b - 17a + b + c = 73put the value of a from equation 1 in equation 2
c = (b + 2) + b − 17c = 2b − 15now, put the value of a and c in equation 3
b + 2 + b + 2b − 15 = 734b − 13 = 734b = 86b = 21.5 inNow, we need to find HJ (a)
a = b + 2a = 21.5 + 2a = 23.5 in[tex]{ \qquad \large \sf {Conclusion} :} [/tex]
HJ = 23.5 inThe function f(x) is shown in the graph.
Which type of function describes f(x)?
O Exponential
O Logarithmic
O Rational
O Polynomial
Answer: Logarithmic
Explanation:
This curve is a reflection of the exponential curve over the line y = x, to show that it is the inverse of exponentials. We use logs to help isolate the exponent among other useful properties.
Which equation does the graph represent?
Answer:
It is the second answer
Step-by-step explanation:
The standard form of an ellipse is
x^2/a^2 + y^2/b^2 or x^2/b^2 + y^2/a^2 = 1
If the x is the main axis we use the first form. If the y is the main axis we use the second form. We will use the second form.
our a is 3 and our b is 4
x^2/3^2 + y^2/4^2
Compute the amount of interest earned in the following simple interest problem. A deposit of $1,295 at 7% for 180 days
(Note: Use 365 days in a year)
since there are 365 days in a year, so 180 days is really just 180/365 of a year, so
[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$1295\\ r=rate\to 7\%\to \frac{7}{100}\dotfill &0.07\\ t=years\to \frac{180}{365}\dotfill &\frac{36}{73} \end{cases} \\\\\\ I = (1295)(0.07)(\frac{36}{73})\implies I\approx 44.7[/tex]
9. will give brainliest
The equation of the parabola with the given vertex and directrix in vertex form is y = (1/16)( x + 5 )² - 9.
Hence, option C is the correct answer.
What is the equation of the parabola?
Given the data in the question;
Vertex of the parabola: ( -5, -9 )h = -5k = -9Directrix of the parabola: y = -13To find the equation, we use the equation of the parabola that opens up or down since the directrix ( y = -13 ) is vertical.
The equation is expressed as;
( x - h )² = 4p( y - k )
First, we find the distance from the focus to the vertex.
|p| is the distance rom the focus to the vertex and from the vertex to the directrix.
p = -9 + 13
p = 4
We substitute the values into the equation;
( x - h )² = 4p( y - k )
( x - (-5) )² = 4(4)( y - (-9) )
( x + 5 )² = 16( y + 9 )
Multiply both side by 1/16
(1/16)( x + 5 )² = y + 9
Make y the subject of the formula
(1/16)( x + 5 )² - 9 = y
y = (1/16)( x + 5 )² - 9
The equation of the parabola with the given vertex and directrix in vertex form is y = (1/16)( x + 5 )² - 9.
Hence, option C is the correct answer.
Learn more about equation of parabola here: https://brainly.com/question/21685473
#SPJ1
find the coefficient of x^5 in the expression ( 1 - 2x) ^6
Answer:
-32x^5
Step-by-step explanation:
using binomial expression we have (1-2x)^6
One angle of a triangle is 30° more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.
Answer: [tex]\boxed{30^{\circ}, 60^{\circ}, 90^{\circ}}[/tex]
Step-by-step explanation:
Let the smallest angle be x.
Then, the middle angle is x+30.
The largest angle is 2x+30.
Angles in a triangle add to 180 degrees, so:
[tex]x+x+30+2x+30=180\\\\4x+60=180\\\\4x=120\\\\x=30[/tex]
So, the angles measure [tex]\boxed{30^{\circ}, 60^{\circ}, 90^{\circ}}[/tex]
The measures of all three angles are 30°,60°, and 90°.
What is meant by triangle?A triangle is a 3-sided polygon occasionally (though not frequently) referred to as the trigon. Every triangle has three sides and three angles, some of which may be the same.
Find the measures of all three angles:
The three angles be a, b and c.
Angle “a" = x
Angle “b" = x + 30°
Angle “c" = “a" + “b"
= x + x + 30°
= 2x + 30°
Total angle in a triangle = 180°
Therefore;
“a" + “b" + “c" = 180°
Births; (x + (x + 30°) + (2x + 30°)) = 180°
So,
4x + 60° = 180°
x = (180° - 60°) ÷ 4
x = 30°
Plugging this value of x into the earlier equations for angles “a,” “b,” and “c."
“a" = x = 30°
“b" = x + 30° = 60°
“c" = 2x + 30° = 90°
The measures of all three angles are 30°,60°, and 90°.
To learn more about triangles, refer to:
https://brainly.com/question/17335144
#SPJ9
f(1)=−6
f(2)=−4
f(n)=f(n−2)+f(n−1)
f(n)=?
The nth term of the sequence is 2n - 8
Equation of a functionThe nth term of an arithmetic progression is expressed as;
Tn = a + (n - 1)d
where
a is the first term
d is the common difference
n is the number of terms
Given the following parameters
a = f(1)=−6
f(2) = −4
Determine the common difference
d = f(2) - f(1)
d = -4 - (-6)
d = -4 + 6
d = 2
Determine the nth term of the sequence
Tn = -6 + (n -1)(2)
Tn = -6+2n-2
Tn = 2n - 8
Hence the nth term of the sequence is 2n - 8
Learn more on nth term of an AP here: https://brainly.com/question/19296260
#SPJ1
By definition, we have
[tex]f(n) = f(n - 1) + f(n - 2)[/tex]
so that by substitution,
[tex]f(n-1) = f(n-2) + f(n-3) \implies f(n) = 2f(n-2) + f(n-3)[/tex]
[tex]f(n-2) = f(n-3) + f(n-4) \implies f(n) = 3f(n-3) + 2f(n-4)[/tex]
[tex]f(n-3) = f(n-4) + f(n-5) \implies f(n) = 5f(n-4) + 3f(n-5)[/tex]
[tex]f(n-4) = f(n-5) + f(n-6) \implies f(n) = 8f(n-5) + 5f(n-6)[/tex]
and so on.
Recall the Fibonacci sequence [tex]F(n)[/tex], whose first several terms for [tex]n\ge1[/tex] are
[tex]\{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots\}[/tex]
Let [tex]F_n[/tex] denote the [tex]n[/tex]-th Fibonacci number. Notice that the coefficients in each successive equation form at least a part of this sequence.
[tex]f(n) = f(n-1) + f(n-2) = F_2f(n-1) + F_1 f(n-2)[/tex]
[tex]f(n) = 2f(n-2) + f(n-3) = F_3 f(n-2) + F_2 f(n-3)[/tex]
[tex]f(n) = 3f(n-3) + 2f(n-4) = F_4 f(n-3) + F_3 f(n-4)[/tex]
[tex]f(n) = 5f(n-4) + 3f(n-5) = F_5 f(n-4) + F_4 f(n-5)[/tex]
[tex]f(n) = 8f(n-5) + 5f(n-6) = F_6 f(n-5) + F_5 f(n-6)[/tex]
and so on. After [tex]k[/tex] iterations of substituting, we would end up with
[tex]f(n) = F_{k+1} f(n - k) + F_k f(n - (k+1))[/tex]
so that after [tex]k=n-2[/tex] iterations,
[tex]f(n) = F_{(n-2)+1} f(n - (n-2)) + F_{n-2} f(n - ((n-2)+1)) \\\\ f(n) = f(2) F_{n-1} + f(1) F_{n-2} \\\\ \boxed{f(n) = -4 F_{n-1} - 6 F_{n-2}}[/tex]
Which radical expression is equivalent to
[tex]~\hspace{7em}\textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-\frac{ n}{ m}} \implies \cfrac{1}{a^{\frac{ n}{ m}}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\[-0.35em] ~\dotfill\\\\ a^{\frac{1}{5}}\implies \sqrt[5]{a^1}\implies \sqrt[5]{a}[/tex]
Simplify 4/16 to lowest terms and find an equivalent fraction that has a denominator of 32
Solve Log6( x) = -2
5. Here are two copies of the same figure. Show two different ways for
finding the area of the shaded region. All angles are right angles.
(Photo below)
The two different ways of finding the area are,
Case 1 = assume horizontal rectangles,
Case 2 = assume vertical rectangles.
the rectangle is a four-sided polygon whose opposites sides are equal and has an angle of 90° between its sides.
Here,
case 1,
As shown in the image
Area = sum of horizontal rectangles
Area = 10 * 3 + 2 * 5 + 2 * 1
Area = 30 + 10 + 2
Area = 42
Case II,
As shown in right figure,
Area of the vertical rectangles
Area = 3 * 5 + 5 * 3 + 2 * 6
Area = 15 + 15 + 12
Area = 42
Here, the area in case 1 is equal to case 2.
Thus, the two different ways of finding the area have shown above.
Learn more about rectangles here:
https://brainly.com/question/16021628
#SPJ1
On Monday, a local hamburger shop sold a combined total of 336 hamburgers and cheeseburgers. The number of cheeseburgers sold was three times the number of hamburgers sold. How many hamburgers were sold on Monday
Answer:
84
Step-by-step explanation:
Let the number of hamburgers be h and the number of cheeseburgers be c.
This means that:
h+c=336c=3hSubstituting c=3h into the first equation, it follows that 4h=336, and thus h=84.
A:HOW MANY DISTINCT ARRANGEMENTS CAN BE MADE WITH THE LETTERS IN THE
WORD CONNECTION?
B: HOW MANY AUTOMOBILE LICENSE PLATES CAN BE MADE IF EACH PLATE
CONTAINS 3 DIFFERENT DIGITS FOLLOWED BY 3 DIFFERENT LETTERS?
Answer:
A:The Reqd. No. of Permutations=3360
Step-by-step explanation:
I couldn’t find the answer to B..sorry
ANSWER FOR BRAINLIEST AND FOR 57 Points If the probability of winning the ball-toss game at a carnival is 20% and the probability of winning the dart game is 15%, what is the probability of winning both? What is the probability of winning either one of these games? Explain your answers.
Answer:
3%, 32%
Step-by-step explanation:
winning 1 game only: two possibilities
a. winning balltoss, losing dart, which is 20%*85% = 17%
b. winning dart, losing ball toss, which is 15%*80% = 12%
so winning 1 game only: 29%
winning both games:
20% * 15% = 3%
winning either one: winning both games+winning 1 game only
29% + 3% = 32%
How does the graph of g(x) = (x − 2)3 + 6 compare to the parent function of f(x) = x3?
g(x) is shifted 2 units to the right and 6 units down.
g(x) is shifted 2 units to the right and 6 units up.
g(x) is shifted 2 units to the left and 6 units down.
g(x) is shifted 6 units to the left and 2 units down.
The relationship of the graph g(x) = (x − 2)^3 + 6 compare to the parent function of f(x) = x^3 is that g(x) is shifted 2 units to the right and 6 units up.
Translation of coordinatesTranslations is a transformation technique that changes the position of an object from one point on the plane to another.
Given the function below
g(x) = (x − 2)^3 + 6
The function compared to f(x) = x^3, shows a translation of f(x) by 2 unit to the right along the horizontal and vertical translation of the function 6 units up
Hence the relationship of the graph g(x) = (x − 2)^3 + 6 compare to the parent function of f(x) = x^3 is that g(x) is shifted 2 units to the right and 6 units up.
Learn more on translation here: https://brainly.com/question/12861087
#SPJ1
Answer:
g(x) is shifted 6 units to the left and 2 units down.
Step-by-step explanation:
took the test
Put y-x=-8 of a line into slope-intercept form, simplifying all fractions.
Answer: [tex]y= x-8[/tex]
Step-by-step explanation:
Slope intercept form has a general formula of [tex]y=mx +b[/tex]m represents the slope of the lineb represents the value of the lines y-intercept the equation must be rearranged into the general formula by isolating for 'y'[tex]y-x=-8[/tex]
to remove the x from the left side of the equation the opposite operation must be done to both sides[tex]y-x+x=-8+x[/tex]
the negative and positive x cancel out on the left side, leaving us with the equation with y by itselfnow you can rearrange to put the equation into [tex]y=mx+b[/tex]Final Answer: [tex]y=x-8[/tex]
A farmer finds there is a linear relationship between the number of bean stalks, n , she plants and the yield, y , each plant produces. When she plants 30 stalks, each plant yields 25 oz of beans. When she plants 32 stalks, each plant produces 24 oz of beans. Find a linear relationship in the form y=mn+b that gives the yield when n stalks are planted.
Answer:
y = -1/2n +40
Step-by-step explanation:
We are given two ordered pairs (stalks, ounces) and asked for the slope-intercept form equation of the line through them.
SlopeThe slope of the desired line can be found from the formula ...
m = (y2 -y1)/(x2 -x1)
For the given points (30, 25) and (32, 24), the slope is ...
m = (24 -25)/(32 -30) = -1/2
Y-interceptThe y-intercept of the desired line can be found from the formula ...
b = y -mn
b = 25 -(-1/2)(30) = 25 +15 = 40
Slope-intercept equationThe slope-intercept equation of a line is ...
y = mn +b . . . . . line with slope m and y-intercept b
y = -1/2n +40 . . . . . . line with slope -1/2 and y-intercept 40
The linear relationship between stalks (n) and yield (y) is ...
y = -1/2n +40
"Solve the following first order differential equation for x(t):
x'=-9tx"
How do I do this?
I'm not sure if the last two apostrophes are part of the quote - "Solve ... " - or if you mean the second derivative [tex]x''[/tex]. I think you mean the first interpretation, but I'll include both cases since they are both solvable.
If the former is correct, separate variables to solve.
[tex]x' = -9tx \implies \dfrac{dx}{dt} = -9tx \implies \dfrac{dx}x = -9t\,dt[/tex]
Integrate both sides to get
[tex]\ln|x| = -\dfrac92 t^2 + C[/tex]
Solve for [tex]x[/tex].
[tex]e^{\ln|x|} = e^{-9/2\,t^2 + C} \implies \boxed{x = Ce^{-9/2\,t^2}}[/tex]
If you meant the latter, then the ODE can be rewritten as
[tex]9t x'' + x' = 0[/tex]
Reduce the order of the equation by substituting [tex]y(t) = x'(t)[/tex] and [tex]y'(t) = x''(t)[/tex].
[tex]9t y' + y = 0[/tex]
Solve for [tex]y'[/tex] and separate variables.
[tex]y' = -\dfrac y{9t} \implies \dfrac{dy}{dt} = -\dfrac y{9t} \implies \dfrac{dy}y = -\dfrac{dt}{9t}[/tex]
Integrate.
[tex]\ln|y| = -\dfrac19 \ln|t| + C[/tex]
Solve for [tex]y[/tex].
[tex]e^{\ln|y|} = e^{-1/9 \,\ln|t| + C} \implies y = Ct^{-1/9}[/tex]
Solve for [tex]x[/tex] by integrating.
[tex]x' = Ct^{-1/9} \implies x = C_1 t^{8/9} + C_2[/tex]
Find the area of the surface given by z = f(x, y) that lies above the region R. f(x, y) = 3 + 4x3/2 R: rectangle with vertices (0, 0), (0, 5), (2, 5), (2, 0)
It looks like the function is
[tex]f(x,y) = 3 + 4x^{3/2}[/tex]
We have
[tex]\dfrac{\partial f}{\partial x} = 6x^{1/2} \implies \left(\dfrac{\partial f}{\partial x}\right)^2 = 36x[/tex]
[tex]\dfrac{\partial f}{\partial y} = \left(\dfrac{\partial f}{\partial y}\right)^2 = 0[/tex]
Then the area of the surface over [tex]R[/tex] is
[tex]\displaystyle \iint_R f(x,y) \, dS = \iint_R \sqrt{1 + 36x + 0} \, dA \\\\ ~~~~~~~~ = \int_0^5 \int_0^2 \sqrt{1+36x} \, dx \, dy \\\\ ~~~~~~~~ = 5 \int_0^2 \sqrt{1+36x} \, dx \\\\ ~~~~~~~~ = \frac5{36} \int_1^{73} \sqrt u \, du \\\\ ~~~~~~~~ = \frac5{36}\cdot \frac23 \left(73^{3/2} - 1^{3/2}\right) = \boxed{\frac5{54} (73^{3/2} - 1)}[/tex]
I honestly need help with these
9. The curve passes through the point (-1, -3), which means
[tex]-3 = a(-1) + \dfrac b{-1} \implies a + b = 3[/tex]
Compute the derivative.
[tex]y = ax + \dfrac bx \implies \dfrac{dy}{dx} = a - \dfrac b{x^2}[/tex]
At the given point, the gradient is -7 so that
[tex]-7 = a - \dfrac b{(-1)^2} \implies a-b = -7[/tex]
Eliminating [tex]b[/tex], we find
[tex](a+b) + (a-b) = 3+(-7) \implies 2a = -4 \implies \boxed{a=-2}[/tex]
Solve for [tex]b[/tex].
[tex]a+b=3 \implies b=3-a \implies \boxed{b = 5}[/tex]
10. Compute the derivative.
[tex]y = \dfrac{x^3}3 - \dfrac{5x^2}2 + 6x - 1 \implies \dfrac{dy}{dx} = x^2 - 5x + 6[/tex]
Solve for [tex]x[/tex] when the gradient is 2.
[tex]x^2 - 5x + 6 = 2[/tex]
[tex]x^2 - 5x + 4 = 0[/tex]
[tex](x - 1) (x - 4) = 0[/tex]
[tex]\implies x=1 \text{ or } x=4[/tex]
Evaluate [tex]y[/tex] at each of these.
[tex]\boxed{x=1} \implies y = \dfrac{1^3}3 - \dfrac{5\cdot1^2}2 + 6\cdot1 - 1 = \boxed{y = \dfrac{17}6}[/tex]
[tex]\boxed{x = 4} \implies y = \dfrac{4^3}3 - \dfrac{5\cdot4^2}2 + 6\cdot4 - 1 \implies \boxed{y = \dfrac{13}3}[/tex]
11. a. Solve for [tex]x[/tex] where both curves meet.
[tex]\dfrac{x^3}3 - 2x^2 - 8x + 5 = x + 5[/tex]
[tex]\dfrac{x^3}3 - 2x^2 - 9x = 0[/tex]
[tex]\dfrac x3 (x^2 - 6x - 27) = 0[/tex]
[tex]\dfrac x3 (x - 9) (x + 3) = 0[/tex]
[tex]\implies x = 0 \text{ or }x = 9 \text{ or } x = -3[/tex]
Evaluate [tex]y[/tex] at each of these.
[tex]A:~~~~ \boxed{x=0} \implies y=0+5 \implies \boxed{y=5}[/tex]
[tex]B:~~~~ \boxed{x=9} \implies y=9+5 \implies \boxed{y=14}[/tex]
[tex]C:~~~~ \boxed{x=-3} \implies y=-3+5 \implies \boxed{y=2}[/tex]
11. b. Compute the derivative for the curve.
[tex]y = \dfrac{x^3}3 - 2x^2 - 8x + 5 \implies \dfrac{dy}{dx} = x^2 - 4x - 8[/tex]
Evaluate the derivative at the [tex]x[/tex]-coordinates of A, B, and C.
[tex]A: ~~~~ x=0 \implies \dfrac{dy}{dx} = 0^2-4\cdot0-8 \implies \boxed{\dfrac{dy}{dx} = -8}[/tex]
[tex]B:~~~~ x=9 \implies \dfrac{dy}{dx} = 9^2-4\cdot9-8 \implies \boxed{\dfrac{dy}{dx} = 37}[/tex]
[tex]C:~~~~ x=-3 \implies \dfrac{dy}{dx} = (-3)^2-4\cdot(-3)-8 \implies \boxed{\dfrac{dy}{dx} = 13}[/tex]
12. a. Compute the derivative.
[tex]y = 4x^3 + 3x^2 - 6x - 1 \implies \boxed{\dfrac{dy}{dx} = 12x^2 + 6x - 6}[/tex]
12. b. By completing the square, we have
[tex]12x^2 + 6x - 6 = 12 \left(x^2 + \dfrac x2\right) - 6 \\\\ ~~~~~~~~ = 12 \left(x^2 + \dfrac x2 + \dfrac1{4^2}\right) - 6 - \dfrac{12}{4^2} \\\\ ~~~~~~~~ = 12 \left(x + \dfrac14\right)^2 - \dfrac{27}4[/tex]
so that
[tex]\dfrac{dy}{dx} = 12 \left(x + \dfrac14\right)^2 - \dfrac{27}4 \ge 0 \\\\ ~~~~ \implies 12 \left(x + \dfrac14\right)^2 \ge \dfrac{27}4 \\\\ ~~~~ \implies \left(x + \dfrac14\right)^2 \ge \dfrac{27}{48} = \dfrac9{16} \\\\ ~~~~ \implies \left|x + \dfrac14\right| \ge \sqrt{\dfrac9{16}} = \dfrac34 \\\\ ~~~~ \implies x+\dfrac14 \ge \dfrac34 \text{ or } -\left(x+\dfrac14\right) \ge \dfrac34 \\\\ ~~~~ \implies \boxed{x \ge \dfrac12 \text{ or } x \le -1}[/tex]
13. a. Compute the derivative.
[tex]y = x^3 + x^2 - 16x - 16 \implies \boxed{\dfrac{dy}{dx} = 3x^2 - 2x - 16}[/tex]
13. b. Complete the square.
[tex]3x^2 - 2x - 16 = 3 \left(x^2 - \dfrac{2x}3\right) - 16 \\\\ ~~~~~~~~ = 3 \left(x^2 - \dfrac{2x}3 + \dfrac1{3^2}\right) - 16 - \dfrac13 \\\\ ~~~~~~~~ = 3 \left(x - \dfrac13\right)^2 - \dfrac{49}3[/tex]
Then
[tex]\dfrac{dy}{dx} = 3 \left(x - \dfrac13\right)^2 - \dfrac{49}3 \le 0 \\\\ ~~~~ \implies 3 \left(x - \dfrac13\right)^2 \le \dfrac{49}3 \\\\ ~~~~ \implies \left(x - \dfrac13\right)^2 \le \dfrac{49}9 \\\\ ~~~~ \implies \left|x - \dfrac13\right| \le \sqrt{\dfrac{49}9} = \dfrac73 \\\\ ~~~~ \implies x - \dfrac13 \le \dfrac73 \text{ or } -\left(x-\dfrac13\right) \le \dfrac73 \\\\ ~~~~ \implies \boxed{x \le 2 \text{ or } x \ge \dfrac83}[/tex]
Solve for x.
31-x=252
Answer: -221
Step-by-step explanation:
31 is smaller than 252. Therefore, since 31 MINUS x equals 252, x needs to be a negative number in order to complete the equation (recall that a negative number times a negative number equals a positive number).
Therefore, if we subtract 31 on both sides, in other words transpose, we get,
-x = 221
The coefficient of -x is -1, however it is not written as it's implied that if there is not written coefficient in from of a variable then the coefficient of the variable is 1 or -1, depending on its sign.
Therefore, dividing -1 on both sides, we get,
x = -221
Hence, the desired answer is -221.
Evaluate f(x)=−4ex−2−4 for x=4. round to the nearest 4 decimal
Answer:
25.2
Step-by-step explanation:
I'll assume you wrote:
[tex]f(x) = 4e^{x-2} - 4[/tex]
So when x = 4:
[tex]4e^{4-2} - 4 = 4e^2 - 4[/tex] ≅ [tex]4\cdot 7.3 - 4 = 25.2[/tex]
6. (a) In the given figure, AD and BC are two straight lines. If ZBAO = 50°, ZABO = 60° and ZPCD = 130° then find the values of x and y. 50 60% B 130
Answer: 70 and 60 degrees
Step-by-step explanation:
Angle AOB = 180 - 50 - 60 = 70 degrees so x is 70 degrees
Angle OCD = 180 - 130 = 50 so y = 180 - 70 - 50 = 60 degrees