Explanation:
Given y defined below:
[tex]y=\frac{2x+1}{x^4+2}[/tex]To find the derivative of y, we use the quotient rule.
[tex]\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}[/tex]In the function:
[tex]\begin{gathered} u=2x+1\implies\frac{du}{dx}=2 \\ v=x^4+2\implies\frac{dv}{dx}=4x^3 \end{gathered}[/tex]Substitute these values into the formula:
[tex]\frac{d}{dx}(\frac{u}{v})=\frac{2(x^4+2)-4x^3(2x+1)}{(x^4+2)^2}[/tex]We then simplify:
[tex]\begin{gathered} \frac{dy}{dx}=\frac{2x^4+4-8x^4-4x^3}{(x^4+2)^2} \\ =\frac{2x^4-8x^4-4+4x^3}{(x^4+2)^2} \\ =\frac{-6x^4+4x^3-4}{(x^4+2)^2} \\ =\frac{-2(3x^4+2x^3-2)}{(x^4+2)^2} \end{gathered}[/tex]Answer:
The derivative dy/dx of the given function is:
[tex]\frac{dy}{dx}=\frac{-2(3x^{4}+2x^{3}-2)}{(x^{4}+2)^{2}}[/tex]Translate this phrase into an algebraic expression.Nine less than the product of 8 and Carlos's savingsUse the variable c to represent Carlos's savings.
It is important to know that less than indicates a subtraction, product indicates multiplication. If C represents Carlos, we can express the situation as follows
[tex]8C-9[/tex]Notice that the number 9 is subtract from the product 8C, that's the correct way to express it.
write an equation of the circle with the given center and radius (-9,3) r=5
write an equation of the circle with the given center and radius
(-9,3) r=5
we know that
the equation of the circle in center radius form is equal to
(x-h)^2+(y-k)^2=r^2
where (h,k) is the center and r is the radius
substitute the given values
(x+9)^2+(y-3)^2=5^2
(x+9)^2+(y-3)^2=25For problems 13-15, complete the chart below and then use the chart to determine the image points.90"(x, y) -(4-22,4)180°(x, y) - 42270"|(x,y) → (1,2-diy)13) Rotate 90" about the origin14) Rotate 180º about the origin15) Rotate 270" about the originM(3, 6) ► M'V(-5, -3) - VA(5, 3) AN(5, 10) - N'W/2. 1) - W'
Given a point (h,k), we can rotate it using the following formulas:
Rotate 90 degrees: (h,k) becomes (k,-h)
Rotate 180 degrees: (h,k) becomes (-h,-k)
Rotate 270 degrees: (h,k) becomes (k,-h)
In this case, we have (for the 90 degrees rotation):
[tex]M(3,6)\text{ }\Rightarrow M^{\prime}(6,-3)\text{ }[/tex]For the 180 degrees rotation:
[tex]V(-5,-3)\Rightarrow V^{\prime}(5,3)[/tex]And finally for the 270 degrees rotation:
[tex]A(5,3)\text{ }\Rightarrow A^{\prime}(3,-5)[/tex]12×20\ 5(0.2) = That's it
Evaluate the value of expression.
[tex]\begin{gathered} \frac{12\times20}{5(0.2)}=\frac{240}{1.0} \\ =240 \end{gathered}[/tex]Answer: 240
given the following matrices, if possible, determine 3A - 3B. if not, state “not possible”
We have:
[tex]3A-3B=3\begin{bmatrix}{6} & {3} & {} \\ {-6} & {-3} & {} \\ {4} & {-2} & {}\end{bmatrix}-3\begin{bmatrix}{8} & {-8} & {4} \\ {7} & {-6} & {-5} \\ {} & {} & \end{bmatrix}[/tex]Multiplying each matrix by 3:
[tex]3A-3B=\begin{bmatrix}{18} & {9} & {} \\ {-18} & {-9} & {} \\ {12} & {-6} & {}\end{bmatrix}-\begin{bmatrix}{24} & {-24} & {12} \\ {21} & {-18} & {-15} \\ {} & {} & \end{bmatrix}[/tex]Then, to facilitate subtraction, we convert each matrix into a 3x3 matrix, filling in the missing column and row with zero. Like this:
[tex]3A-3B=\begin{bmatrix}{18} & {9} & {0} \\ {-18} & {-9} & {0} \\ {12} & {-6} & {0}\end{bmatrix}-\begin{bmatrix}{24} & {-24} & {12} \\ {21} & {-18} & {-15} \\ {0} & {0} & {0}\end{bmatrix}[/tex]So, we do the subtraction of each number that makes up the matrix:
[tex]\begin{gathered} 3A-3B=\begin{bmatrix}{18-24} & {9-(-24)} & {0-12} \\ {-18-21} & {-9-(-18)} & {0-(-15)} \\ {12-0} & {-6-0} & {0-0}\end{bmatrix} \\ 3A-3B=\begin{bmatrix}{-6} & {33} & {-12} \\ {-39} & {9} & {15} \\ {12} & {-6} & {0}\end{bmatrix} \end{gathered}[/tex]Answer:
[tex]\begin{bmatrix}{-6} & {33} & {-12} \\ {-39} & {9} & {15} \\ {12} & {-6} & {0}\end{bmatrix}[/tex]Diego‘s doctor has recommended that his daily diet should include three vegetables, two fruits, and two whole grains. At the grocery store Diego has a choice of 9 vegetables 9 fruits and 11 whole grains. In how many ways can he get his daily requirements if he doesn’t like to eat two servings of the same thing in one day
Step 1
Given;
Step 2
[tex]\begin{gathered} 9C_3\times11C_2\times9C_2 \\ nCr=\frac{n!}{r!\left(n-r\right)!} \end{gathered}[/tex][tex]84\times55\times36=166320[/tex]Since he does not like to eat two servings of the same thing in one day, then the answer will be;
[tex]166320[/tex]Answer;
[tex]166320\text{ ways}[/tex]13.2 m 3m 17m What is the total surface area of the rectangular prism in square meters?
Answer: Total surface area = 642.96 square meters
Explanation:
The completed prism is shown in the diagram below
The total surface area is the sum of the area of each face. the faces are in pairs. Thus,
Total surface area = 2(3 x 17.4) + 2(17.4 x 13.2) + 2(3 x 13.2) =
Total surface area = 104.4 + 459.36 + 79.2
Total surface area = 642.96 square meters
You need to purchase all the ingredients in order to make all 3 recipes. Before you go shopping you will first check which ingredients you have in your cabinets. When taking inventory, you found that you already have 2 eggs How many more eggs are needed in order to make all 3 recipes?
You need:
1 egg for Banana bread
1 large egg for sugar cookies
2 eggs for chocolate chip cookies
Total eggs = 4
Since you already have 2 eggs, you will need two more eggs in order to make all 3 recipes
The number of bacteria in a refrigerated food product is given by N(T) = 27T^2 - 155T + 66, 6 < T < 36, where T is the temperature of the food.When the Food is removed from the refrigerator, the tempersture is given by T(t) = 6t + 1.7, where t is the time in hours.Find the composite function N(T(t)): N(T(t)) = 927t^2 - 379.2t - 119.47 (Already solved, don't need help with this question)Find the time when the bacteria count reaches 26087. Time needed = ____ hours(Not solved, need help!)
Given the functions:
[tex]\begin{gathered} N(T)=27T^2-155T+66 \\ \\ T(t)=6t+1.7 \end{gathered}[/tex]Where:
T is the temperature of the food.
t is the time in hours.
Let's solve for the following:
• (a). Find the composite N(T(t).
To find the composite function, we have:
N(T(t)) = N(6t + 1.7)
Substitute (6t+1.7) for T in N(T) and solve for N(6t + 1.7).
We have:
[tex]\begin{gathered} N(6t+1.7)=27(6t+1.7)^2-155(6t+1.7)+66 \\ \\ =27(6t+1.7)(6t+1.7)-155(6t)-155(1.7)+66 \\ \\ =27(6t(6t+1.7)+1.7(6t_{}+1.7))-930t-263.5+66 \\ \\ =27(6t(6t)+6t(1.7)+1.7(6t)+1.7(1.7))-930t-197.5 \end{gathered}[/tex]Solving further:
[tex]\begin{gathered} =(27(36t^2)+27(20.4t)+27\cdot2.89)-930t-197.5 \\ \\ =972t^2+550.8t+78.03-930t-197.5 \\ \\ =972t^2+550.8t-930t+78.03-197.50 \\ \\ =972t^2-379.2t-119.47 \end{gathered}[/tex]Therefore, the composite function is:
[tex]N(T(t))=972t^2-379.2t-119.47[/tex]• (b). Find the time when the bacteria count reaches 26087.
Substitute 26087 for N(T(t)) and solve for t.
We have:
[tex]26087=972t^2-379.2t-119.47[/tex]Equate to zero.
Subtract 26087 from both sides:
[tex]\begin{gathered} 972t^2-379.2t-119.47-260.87=26087-26087 \\ \\ 972t^2-379.2t-26206.47=0 \end{gathered}[/tex]Solve using quadratic formula:
[tex]t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]Apply the standard quadratic formula to find the values of a, b, and c:
[tex]\begin{gathered} ax^2+bx+c=0 \\ \\ 972t^2-379.2t-26206.47=0 \\ \\ \text{Factor out 0.03 from each term:} \\ 0.03(32400t^2-12640t-873549)=0 \\ \\ 32400t^2-12640t-873549=0 \end{gathered}[/tex]Thus, we have:
a = 32400
b = -12640
c = -873429
Input the values into the quadratic formula for solve for t:
[tex]t=\frac{-(-12640)\pm\sqrt[]{113371720000_{}}}{2(32400)}[/tex]Solving further:
[tex]\begin{gathered} t=\frac{12640\pm200\sqrt[]{2834293}}{64800} \\ \\ t=\frac{316\pm5\sqrt[]{2834293}}{1620} \\ \\ t=\frac{316-5\sqrt[]{2834293}}{1620},\frac{316+5\sqrt[]{2834293}}{1620} \\ \\ t=-5.001,\text{ 5.}39 \end{gathered}[/tex]We have the values:
t = -5.001
t = 5.39
Since the time cannot be negative, let's use the positive value.
Therefore, the time needed is 5.39 hours.
ANSWER:
5.39 hours.
Does this function have an output? If not, explain why.
Given function is,
[tex]f(x)=\sqrt[]{x}+1[/tex]So the value for x subsituted here is -2. so we get,
[tex]\begin{gathered} f(-2)=\sqrt[]{-2}+1 \\ f(-2)=i\sqrt[]{2}+1 \end{gathered}[/tex]So the requred answer is the function cannot have a real value, because it has a negative sign inside the root. so it is not possible for this function to have an real value output.
But the function can have an output in Imaginary value ( i ).
Question 13 of 28Which of the following represents the ratio of the long leg to the short leg inthe right triangle shown below?30O A. 1:4√3OB. √3:1O C. 1:2O D. 2:160°
Solution:
Given a right triangle;
Where the long leg, x, is, the side opposite the angle 60°
And the short leg, y, is the side opposite angle 30°
To find the ratio of the long leg to short leg,
[tex]\begin{gathered} For\text{ long leg} \\ \sin\theta=\frac{Opposite}{Hypotenuse} \\ Where \\ \theta=60\degree \\ \sin60\degree=\frac{x}{Hypotenuse} \\ \frac{\sqrt{3}}{2}=\frac{x}{Hypotenuse} \\ Hypotenuse=\frac{2x}{\sqrt{3}} \end{gathered}[/tex]For the short leg
[tex]\begin{gathered} \cos\theta=\frac{Adjacent}{Hypotenuse} \\ \theta=60\degree \\ \cos60\degree=\frac{y}{Hypotenuse} \\ Where \\ \cos60\degree=\frac{1}{2} \\ \frac{1}{2}=\frac{y}{Hypotenuse} \\ Hypotenuse=2y \end{gathered}[/tex]The ratio of the long and short leg will be
[tex]\begin{gathered} \frac{2x}{\sqrt{3}}=2y \\ \frac{2x}{2y}=\frac{\sqrt{3}}{1} \\ \frac{x}{y}=\frac{\sqrt{3}}{1} \\ x:y=\sqrt{3}:1 \end{gathered}[/tex]Hence, the answer is B.
olve for xx. Round to the nearest tenth of a degree, if necessary
Answer:
x = 36.3°
Explanation:
Taking into account the figure, we know that the opposite side of angle x is 3.9 and the adjacent side is 5.3, then using tangent, we get
[tex]\begin{gathered} \tan x=\frac{Opposite}{Adjacent} \\ \tan x=\frac{3.9}{5.3} \\ \tan x=0.75 \end{gathered}[/tex]Then, we can solve for x using the inverse function of tangent
[tex]\begin{gathered} x=\tan ^{-1}(0.75) \\ x=36.3\degree \end{gathered}[/tex]So, x = 36.3°
For this equation, tell whether it is always true, sometimes true, or nevertrue.
To answer the question we make the sum on the left side of the equation:
[tex]\begin{gathered} \frac{1}{2}y+5=\frac{y+10}{2} \\ \frac{y}{2}+5=\frac{y+10}{2} \\ \frac{y}{2}+\frac{10}{2}=\frac{y+10}{2} \\ \frac{y+10}{2}=\frac{y+10}{2} \end{gathered}[/tex]Since both sides are exactly the same. this equation is always true.
Find the length of KB¯¯¯¯¯¯¯¯+AK¯¯¯¯¯¯¯¯. A. 28B. 35C. 21D. 33
For this problem, we are given a circle with two intersecting chords. We need to determine the sum of the segments KB and AK.
We can use the following identity to solve for x.
[tex]DK\cdot KB=AK\cdot KC[/tex]We have:
[tex]\begin{gathered} 12\cdot(4x+1)=14\cdot(3+3x)\\ \\ 48x+12=42+42x\\ \\ 48x-42x=42-12\\ \\ 6x=30\\ \\ x=5 \end{gathered}[/tex]Now we need to compute the distances:
[tex]KB+AK=14+4\cdot(5)+1=35[/tex]The correct answer is B.
On a piece of paper, graph y = x2 + 4x - 5 and identify the y-intercept. Thendetermine which answer choice matches the graph that you drew andcorrectly identifies the y-intercept.
We have
[tex]y=x^2+4x-5[/tex]First, let's graph some points in order to graph the function
x=-4
y=(-4)^2+4(-4)-5=-5
point (-4,-5)
x=-2
y=-9
point (-2,-9)
x=0
y=-5
point (0,-5)
x=2
y=7
point (2,7)
x=4
then we graph the points, and we obtain the next graph
the y-intercept is the value of the coordinate y when the x-coordinate is zero, as we can see in the image the y-intercept is in the point in purple (0,-5)
the y-intercept is -5
Can you help me with the letter a and b please
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given function
[tex]c(t)=108e^{-0.08t}+75[/tex]A letter carrier has 120 letters to deliver. Today they delivered d letters before lunch. Write an expression to show how many letters the carrier still has to deliver before the end of the day.
The constant is 120
The variable is d
The expression is 120 - d
can you please help with these two questions 1). s/6-5= -8. 2). w/8 - 15 =22
Select all the equation that are equivalent.A. 52 = 8n + 4B. 4(2n + 1) = 52 C. 4n = 48E. n=6
A.
[tex]\begin{gathered} 52=8(6)+4 \\ 52=48+4 \\ 52=52 \end{gathered}[/tex]answer A is equivalent
B.
[tex]\begin{gathered} 4(2n+1)=52 \\ 4(2(6)+1)=52 \\ 4(12+1)=52 \\ 4(13)=52 \\ 52=52 \end{gathered}[/tex]B is equivalent
therefore A and B are equivalent
What is the value of the expression 268.75 2.5
In which table is y a linear function of x?42
Let's first solve the given inequality
[tex]c-12>-16[/tex]now we can add 12 in both sides of the inequality, obtaining
[tex]c>-4[/tex]then the graph of the solution will be the first one of the options.
how to solve 165% of 10
To calculate the 165% of 10 we have to multiply 10 by the percentage in decimal form ( divided by 100).
165 %= 165/100 = 1.65 (decimal form)
10 x 1.65 = 16.5
165% of 10 is 16.5
the slope of a line that passes through points (2,6) and ( 3,4) is 2/5. true or false
Given:
Point 1: x1, y1 = -3, 4
Point 2: x2, y2 = 2, 6
Let's determine the slope using the following formula:
[tex]\text{ Slope (m) = }\frac{y_2-y_1}{x_2-x_1_{}}[/tex]We get,
[tex]\text{ Slope (m) = }\frac{y_2-y_1}{x_2-x_1}[/tex][tex]\text{ Slope (m) = }\frac{6-4}{2-(-3)}[/tex][tex]\text{ = }\frac{6\text{ - 4}}{2\text{ + 3}}[/tex][tex]\text{Slope (m) = }\frac{2}{5}[/tex]The slope is 2/5.
Therefore, the answer is False
identify the domain and rangey=-5 + 4 √x-2
Given:
[tex]\begin{gathered} y=-5+4\sqrt[]{x+2} \\ \\ \end{gathered}[/tex]The domain will be all set of possible x values
The range will be all set of possible y values.
Let's find the domain,
Since any value of x less than -2 will make the equation undefined, the value of x must be greater or equal to -2.
Thus:
[tex]undefined[/tex]In the triangle below, suppose that m ∠K=(x+6), m ∠L= (5x-2), and m ∠M=(2x)º.Find the degree measure of each angle in the triangle.
m ∠K= 28º
m ∠L= 108º
m∠M= 44º
Explanation:m ∠K=(x+6), m ∠L= (5x-2), and m∠M=(2x)º
m ∠K + m ∠L + m∠M = 180º (sum of angles in a triangle)
(x+6) + (5x-2) + (2x) = 180
collect like terms:
x + 5x + 2x + 6 - 2 = 180
8x + 4 = 180
8x = 180 -4
8x = 176
Divide both sides by 8:
8x/8 = 176/8
x = 22
m ∠K=(x+6) = 22 + 6
m ∠K= 28º
m ∠L= (5x-2) = 5(22) - 2
m ∠L= 108º
m∠M=(2x) = 2(22)
m∠M= 44º
Find the range and standard deviation of the set of data190, 191, 192. 193, 194. 195, 196
The range is given for the comparison between the smallest and largest value from the set of data, in this case:
[tex]\text{Range}=196-190=6[/tex]The standar deviation is given for the following formula:
[tex]S=\sqrt[]{\frac{\sum ^n_{i=1}(x_i-\bar{x})^2}{n-1}}[/tex]In this case the standart deviation is 2.
Hello, I need some help with this precalculus question for my homework, please HW Q9
We are given the following equation:
[tex]3^{x-3}=81[/tex]First, we will use the following property of exponents:
[tex]a^{x+y}=a^xa^y[/tex]Applying the property we get:
[tex]3^x3^{-3}=81[/tex]Now, we divide both sides by 3^-3:
[tex]3^x=\frac{81}{3^{-3}}[/tex]Now, we use the following property of exponentials:
[tex]a^{-x}=\frac{1}{a^x}[/tex]Applying the property we get:
[tex]3^x=(81)(3^3)[/tex]Solving the product of the right side:
[tex]3^x=2187[/tex]Now, we take the natural logarithm to both sides:
[tex]ln3^x=ln2187[/tex]Now, we use the following property of logarithms:
[tex]lnx^y=ylnx[/tex]Applying the property we get:
[tex]xln3=ln2187[/tex]Dividing both sies by ln3 we get:
[tex]x=\frac{ln2187}{ln3}[/tex]Solving the operations:
[tex]x=7[/tex]Therefore, the exact solution of the equation is 7.
The solution rounded to three decimal places is 7.000
Please help me on 1I’m confused Please show work so I can understand
Answer:
[tex]\log _52+2\log _5x^{}-3\log y[/tex]Explanation:
Given the logarithm expression:
[tex]\log _5\mleft(\frac{2 x^{2}}{y^{3}}\mright)[/tex]To expand the expression, follow the steps below:
Step 1: Apply the division law of logarithm below. That is, the log of a quotient is the difference between the logs. Therefore:
[tex]\begin{gathered} \log (\frac{A}{B})=\log (A)-\log (B) \\ \implies\log _5\mleft(\frac{2 x^{2}}{y^{3}}\mright)=\log _5(2x^2)-\log (y^3) \end{gathered}[/tex]Step 2: Similarly, by the multiplication law, the log of a product is the sum of the logs.
[tex]\begin{gathered} \log (AB)=\log (A)+\log (B) \\ \log _5(2x^2)=\log _52+\log _5x^2 \\ \implies\log _5(2x^2)-\log (y^3)=\log _52+\log _5x^2-\log (y^3) \end{gathered}[/tex]Step 3: We apply the index law of logarithm.
If the number whose logarithm we are looking for has an index (or power), we can write the index as a product.
[tex]\log x^n=n\log x[/tex]So, we have that:
[tex]\implies\log _5(2x^2)-\log (y^3)=\log _52+2\log _5x^{}-3\log y[/tex]Thus, the expanded form of the given expression is:
[tex]\log _5\mleft(\frac{2 x^{2}}{y^{3}}\mright)=\log _52+2\log _5x^{}-3\log y[/tex]
help me g ...........
Answer:
Equation : 0.05 b = 4.60
b = 92
Explanation:
Let us call b the cost of the haircut. We then kn
please help ASAP!!!!!