Note that [tex]+\vee-[/tex] stands for plus or minus.
For the quadratic equation of the form [tex]ax^2+bx+c=0[/tex] the solutions are [tex]x_{1,2}=\dfrac{-b+\vee-\sqrt{b^2-4ac}}{2a}[/tex] that means that for [tex]a=4, b=-10, c=5\Longrightarrow x_{1,2}=\dfrac{-(-10)+\vee-\sqrt{(-10)^2-4\cdot4\cdot5}}{2\cdot4}[/tex] this simplifies to [tex]\boxed{x_1=\dfrac{5+\sqrt{5}}{4}}, \boxed{x_2=\dfrac{5-\sqrt{5}}{4}}[/tex]
Hope this helps.
Answer:
[tex]\large\boxed{x=\dfrac{5-\sqrt5}{4},\ x=\dfrac{5+\sqrt5}{4}}[/tex]
Step-by-step explanation:
[tex]\text{The quadratic formula for}\ ax^2+bx+c=0\\\\\text{if}\ b^2-4ac<0,\ \text{then the equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then the equation has one solution:}\ x=\dfrac{-b}{2a}\\\\\text{if}\ b^2-4ac,\ ,\ \text{then the equation has two solutions:}\ x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\==========================================[/tex]
[tex]\text{We have the equation:}\ 4x^2-10x+5=0\\\\a=4,\ b=-10,\ c=5\\\\b^2-4ac=(-10)^2-4(4)(5)=100-80=20>0\\\\x=\dfrac{-(-10)\pm\sqrt{20}}{2(4)}=\dfrac{10\pm\sqrt{4\cdot5}}{8}=\dfrac{10\pm\sqrt4\cdot\sqrt5}{8}=\dfrac{10\pm2\sqrt5}{8}\\\\=\dfrac{2(5\pm\sqrt5)}{8}=\dfrac{5\pm\sqrt5}{4}[/tex]
Write the equation 9y = 12x + 0.2 in standard form. Identify A, B, and C.
Question 19 options:
45x – 60y = 1 where A = 45, B = –60, and C = –1
45x – 1y = 1 where A = 45, B = –60, and C = 1
60x + 45y = –1 where A = 60, B = 45, and C = 1
60x – 45y = –1 where A = 60, B = –45, and C = –1
Answer: D) 60x-45y = - 1 where A = 60 , B = -45 , and c = - 1
Step-by-step explanation: Clear it : 45y=60x+1
Step 2: Isolate the constant on one side: ( -1 = -45y +60x)
step 3: A= 60 , B= - 45 , C= - 1
Answer:
60x-45y = - 1 where A = 60 , B = -45 , and c = - 1
Step-by-step explanation:
In a class of 30 students 15 know Italian, 10 know French, and 3 know both languages. The rest of the students learn Spanish. How many students learn Spanish?
8 students know Spanish since 12 students know French and 7 students know Italian
Answer:
8 students learn Spanish.
Step-by-step explanation:
In a class total number of students are 30.
Students who know Italian = 15
Students who know French = 10
Students who know both = 3
Rest all students know Spanish.
By Venn diagram attached number of students who know Spanish
= [ 30 - (12 + 3 + 7 )]
= 30 - (22)
= 8 students.
8 students learn Spanish.
If f = {(4, 2), (6, 1), (8, 4), (10, 2), (12, 5)}, what is the range
Answer:
Range = {2,1,4,5}
Step-by-step explanation:
When a function is given in the form if a relation. i.e. ordered pairs.
When the function is given in the form of ordered pairs then the set of first elements i.e. x-coordinates of all ordered pairs forms domain while the set of second elements i.e. y-coordinate of all ordered pair is called range.
So in the given function:
Range = {2,1,4,5}
The repeating values are only written once..
Use substitution to solve each system of equations. y = 4x + 22 4x – 6y = –32
(–5, 2)
(2, –5)
(–8, 1)
(4, 7)
Answer:
(-5,2)
Step-by-step explanation:
The given system is
1st equation: y = 4x + 22
2nd equation: 4x – 6y = –32
We plug in the first equation into the second equation to obtain:
4x – 6(4x + 22) = –32
We expand the parenthesis to obtain:
4x – 24x -132= –32
Group similar terms;
4x – 24x = –32+132
Combine similar terms
-20x =100
Divide both sides by -20
x =-5
Put x=-5 into the 1st equation
y = 4(-5) + 22
y=-20+22
y=2
The solution is:
(-5,2)
A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 220 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?
Answer:
* The largest area can be enclosed is 6050 feet²
Step-by-step explanation:
* Lets explain the situation to solve the problem
- There is a rectangular parking
- The parking will surrounded by fencing from three sides only
- The length of fencing is 220 feet
- Lets consider the width of the rectangle is x and the length of it is y
- The side along the street will not fence
* Lets put all of these data in equation
∵ The width of the parking is x
∵ The length of the parking is y
- He will not fence the side along the street
∴ The perimeter of the parking = x + y + x
∴ The perimeter of the parking = 2x + y
- The length of the fencing = the perimeter of the park
∵ The length of the fencing = 220 feet
∵ The perimeter of the parking = 2x + y
∴ 2x + y = 220 ⇒ (1)
- Lets find the area of the parking
∵ The area of any rectangle is length × width
∵ The width of the rectangle is x
∵ The length of the rectangle is y
∴ The area of the parking (A) = x × y
∴ The area of the parking = xy ⇒ (2)
- Lets find the value of y from equation (1) and substitute this value
in equation (2)
∵ 2x + y = 220 ⇒ subtract 2x from both sides
∴ y = 220 - 2x
- Substitute this value in equation (2)
∵ A = xy
∴ A = x(220 - 2x) ⇒ open the bracket
∴ A = 220x - 2x²
- To find the largest area differentiate the area with respect to x
and equate the result by 0 to find x which gives the largest area
∵ A = 220x - 2x²
- Lets remember the differentiation rules
# If y = a x^n, where a is the coefficient of x then dy/dx = (an) x^(n-1)
# If y = ax, then dy/dx = a
# If y = a, where a is constant then dy/dx = 0
∴ dA/dx = 220 - 2(2) x^(2-1)
∴ dA/dx = 220 - 4x
- Put dA/dx = 0 ⇒ for largest area
∵ dA/dx = 0
∴ 220 - 4x = 0 ⇒ add 4x to both sides
∴ 220 = 4x ⇒ divide both sides by 4
∴ 55 = x
* The width of the parking is 55 feet
- Substitute this value of x in the equation of the area to find the
largest area
∵ A = 220x - 2x²
∵ x = 55
∴ A = 220(55) - 2(55)² = 12100 - 6050 = 6050 feet²
* The largest area can be enclosed is 6050 feet²
To find the largest area that can be enclosed by the given amount of fencing, we can use the concept of optimization and solve for a rectangular lot's dimensions that maximize the area. The largest area that can be enclosed is 6050 square feet.
Explanation:
To find the largest area that can be enclosed, we can use the concept of optimization. Let's assume the length of the rectangular lot is x feet. The remaining length, which is not fenced, will be 220 - 2x feet. The width of the lot will be y feet. So we have 2x + y = 220. To find the largest area, we need to express the area in terms of a single variable. A = xy, substitute y = 220 - 2x. So A = x(220 - 2x) = 220x - 2x^2. To find the maximum value of A, we can find the vertex of the parabola, which corresponds to the maximum. The x-coordinate of the vertex is -b/2a. In this case, a = -2, b = 220, so x = -220/(-2*2) = 55.Plug this value of x into the equation y = 220 - 2x. y = 220 - 2(55) = 110. So the dimensions of the lot that will enclose the largest area are 55 feet by 110 feet. Substituting these values into the area formula A = xy, we get A = 55 * 110 = 6050 square feet.
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Eric deposited $9,033.00 into a new savings account that earns interest compounded monthly. After 11 months, the balance in the account was $10,230.00. What was the interest rate on the account?
Round your answer to the nearest tenth of a percent.
Dang that’s tuff I feel bad
Willie has 4 baseball caps. Two of the caps are blue. One of the caps is red and one is green. What fraction of the caps is blue?
Answer:
2/4
when simplified = 1/2
Your answer is 1/2
Step-by-step explanation:
Amt. of Baseball caps = 4
Blue caps = 2
which can be written as 2/4
when simplified 2/4 ÷ 2/2 = 1/2
Willie has 4 baseball caps, of which 2 are blue. To find the fraction of caps that are blue, divide the number of blue caps (2) by the total number of caps (4), resulting in a fraction of 1/2.
Explanation:Willie has 4 baseball caps in total, and 2 of those caps are blue. To determine what fraction of the caps is blue, you divide the number of blue caps by the total number of caps.
So, the calculation would be:
Number of blue caps = 2Total number of caps = 4Fraction of caps that are blue = Number of blue caps ÷ Total number of caps = 2 ÷ 4 = 1/2Therefore, the fraction of Willie's caps that are blue is 1/2, which means half of the caps are blue.
Can someone help me with this math question?
Answer:
see attachment
Step-by-step explanation:
It is convenient to use a spreadsheet for this purpose.
The first row of numbers is constant at $45, as there is no daily charge associated with that payment method.
The second row of numbers uses the formula ...
cost = $12 + $4×(number of days)
The third row of numbers uses the formula ...
cost = $6×(number of days)
___
Of course, you can use these formulas to fill in the numbers by hand. For example, for 15 days, the charges are ...
Early Pay: $45 (no calculation necessary)Deposit Plus: $12 + $4×15 = $12 +60 = $72Daily Pay: $6×15 = $90f(x)=4x2+8x-9 find and simplify f(2+x)
a. 23+x
b. 4x^2+8x-11
c. 4x^2+2x+23
d. 4x^2+24x+23
Answer:
d. 4x^2 +24x +23
Step-by-step explanation:
Evaluating polynomials is sometimes easier when they are written in Horner form:
f(x) = (4x +8)x -9
Substituting (x+2) for x, we have ...
f(x+2) = (4(x+2)+8)(x+2) -9
= (4x +16)(x +2) -9
= 4x^2 +24x +32 -9
= 4x^2 +24x +23 . . . . . matches choice D
_____
Alternative method
You can observe that the answer choices differ in the coefficient of the x-term. So, to make the correct selection, you only need to find the coefficient of the x-term. That will be the sum of coefficients of the x-terms in 4(x+2)^2 and 8(x+2). Those x-terms are 4·4x and 8x, so have a sum of (16+8)x = 24x. This matches choice D.
Solve for x. x2 - 2x - 24 = 0 A. -4, -6 B. -4, 6 C. 2, -6 D. 4, 6 Reset Next
Answer:
B. -4, 6
Step-by-step explanation:
Given the quadratic equation;
x^2 - 2x - 24 = 0
we can determine the solution by first factoring the expression on the left hand side. We determine two numbers whose product is -24 and sum -2. By trial and error the two numbers are found to be;
-6 and 4
We replace the middle term, -2x, with these two values;
x^2 + 4x -6x -24 = 0
x(x+4) -6(x+4) = 0
(x-6)(x+4) = 0
x-6 = 0 or
x + 4 = 0
x = 6 or x = -4
Answer: the answer is B
Step-by-step explanation: -4 & 6 are both zeros of the function .
Mike wants to redesign a box. Currently, it’s length is 20 cm, it’s width is 30 cm and it’s height is 40cm. he wants to keep the volume and the length unchanged and increase the height by 25 percent. What will be the new width of the box?
Answer:
24 cm
Step-by-step explanation:
The product of height and width will remain the same, so the new width (w) will be given by ...
w·(new height) = (old width)·(old height)
w·(1.25·40 cm) = (30 cm)(40 cm)
w = (30 cm)/1.25 = 24 cm . . . . . . . divide by 1.25·40 cm and simplify
_____
This derives from the fact that volume and length are unchanged. The formula for the volume in terms of length, width, and height is ...
V = LWH
Then the product of W and H is the constant ...
V/L = WH . . . . . divide by L
For our purpose, we only need to know that V and L are unchanged, so the product WH is unchanged. We don't need to know their values.
Of course, increasing the height by 25% is equivalent to multiplying it by 1.25:
H + 25/100·H = H·(1 + 0.25) = 1.25H
A sample of 64 observations is selected from a normal population. The sample mean is 215, and the population standard deviation is 15. Conduct the following test of hypothesis using the 0.025 significance level. H0: μ ≥ 220 H1: μ < 220 Is this a one- or two-tailed test? One-tailed test Two-tailed test What is the decision rule? (Negative amount should be indicated by a minus sign. Round your answer to 2 decimal places.) What is the value of the test statistic? (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.) What is your decision regarding H0? Reject Do not reject What is the p-value? (Round your answer to 4 decimal places.) rev: 10_28_2017_QC_CS-107404 Next Visit question mapQuestion 2 of 4 Total 2 of 4 Prev
Answer:
15 million
Step-by-step explanation:
This is a one-tailed test with a significance level of 0.025. The test statistic is -1.83, and we reject the null hypothesis. The p-value is approximately 0.0344.
Explanation:This is a one-tailed test because the alternative hypothesis (H1) is specifying a less than condition (<) for the population mean.
The decision rule for a one-tailed test with a significance level of 0.025 is to reject the null hypothesis (H0) if the test statistic is less than the critical value.
The test statistic is calculated by subtracting the hypothesized population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. In this case, the test statistic is [(215 - 220) / (15 / sqrt(64))] = -1.83 (rounded to 3 decimal places).
In order to make a decision regarding H0, we compare the test statistic with the critical value. If the test statistic is less than the critical value, we reject H0. Otherwise, if the test statistic is greater than or equal to the critical value, we fail to reject H0. In this case, -1.83 is less than the critical value, so we reject H0.
The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming that the null hypothesis is true. To find the p-value, we look up the test statistic in the standard normal distribution table. In this case, the p-value is approximately 0.0344 (rounded to 4 decimal places).
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What is the value of the expression |a + b| + |c| when a = –3, b = –7, and c = –15?
Answer:
Step-by-step explanation:
Formula
abs(a + b) + abs(c)
Givens
a = - 3
b = - 7
c = - 15
Solution
abs(-3 - 7) + abs(-15)
abs(-10) + abs(-15)
10 + 15
25
Please help me last question
As you can see there are two triangles but it can be done calculating just for one. First we must understand how formula for area of triangle works.
[tex]A=\frac{1}{2}bh[/tex]
Where [tex]b[/tex] represents base (hypotenuse) and [tex]h[/tex] as height of the triangle.
We know that:
[tex]
b=13cm \\
h=4cm
[/tex]
Using this data we fill the formula.
[tex]A=\frac{1}{2}\cdot13\cdot4=\frac{13\cdot4}{2}=13\cdot2=\boxed{26cm^2}[/tex]
Hope this helps.
r3t40
Answer:
Area of the triangle = 26 cm²
Step-by-step explanation:
The given triangle has the measure of height h = 4 cm
and base of the triangle = 13 cm
We know the formula of the area of a triangle = [tex]\frac{1}{2}(Base)(height)[/tex]
By putting the values in the formula
Area of the triangle = [tex]\frac{1}{2}(4)(13)[/tex]
= 2×13
= 26 cm²
Therefore, area of the given triangle is 26 cm².
Solve for x. Geometry plz help
- The solution/answer is 10.
The two angles are Supplementary angles and need to eaual 1980 degrees.
x-2 + 5x +2 = 180
Simplify:
6x = 180
Divide both sides by 6:
x = 180 /6
x = 30
You find an old bathroom scale at a garage sale on your way home from getting a physical exam from your doctor. You step on the scale, and it reads 135 lb. You step off and step back on, and it reads 134 lb. You do this three more times and get readings of 135 lb, 136 lb, and 135 lb. a. What is the precision of this old bathroom scale? Would you consider this adequate precision for the type of measurement you are making? b. The much more carefully constructed and better-maintained scale at the doctor's office reads 126 lb. Assuming that you are wearing the same clothes that you wore when the doctor weighed you, do you think the accuracy of the old bathroom scale is high or low?
the scale has a persicion of 2 and it reads high.
1. 2 is the precision of this old bathroom scale.
2. Since 135 lbs of the old scale is higher than 126 lbs of a better-maintained scale at the doctor's office.
Given that,
An old bathroom scale,
You step on the scale, and it reads 135 lb. You step off and step back on, and it reads 134 lb. You do this three more times and get readings of 135 lb, 136 lb, and 135 lb.
In mathematics, it deals with numbers of operations according to the statements.
Here,
a). What is the precision of this old bathroom scale,
= higher reading - the lower reading
= 136 - 134
= 2
b. The much more carefully constructed and better-maintained scale at the doctor's office reads 126 lb.
Since measured weight by the old scale is 135 lbs which is higher than 126 lbs measured by the scale at the doctor's office.
Thus,
1. 2 is the precision of this old bathroom scale.
2. Since 135 lbs of the old scale is higher than 126 lbs of a better-maintained scale at the doctor's office.
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Please help me simplified this
For this case we have that by definition, the perimeter of the quadrilateral shown is given by the sum of its sides:
Let "p" be the perimeter of the quadrilateral, then:
[tex]p = 7 + y + 7 + x\\p = 7 + 7 + x + y\\p = 14 + x + y[/tex]
So, the perimeter of the figure is: [tex]14 + x + y[/tex]
Answer:
[tex]p = 14 + x + y[/tex]
Answer:
perimeter = x + y + 14
Step-by-step explanation:
The perimeter is the sum of the side lengths:
perimeter = x + 7 + y + 7
The constants can be combined to give ...
perimeter = x + y + 14
___
An "equation" will have an equal sign somewhere. Since you want to find the perimeter, it makes sense for the equation to show you how to find the perimeter. (I have used "perimeter" to represent the perimeter. It is often represented using the letter P. Of course, you can choose any variable or other representation you like.)
A spinner is divided into four equal sections that are numbered 2, 3, 4, and 9. The spinner is spun twice. How many outcomes have a product less than 20 and contain at least one even number?
Answer with Step-by-step explanation:
On spinning the spinner twice,we have 16 different outcomes:.
We write the outcomes with their product:
Product
2 2 4
2 3 6
2 4 8
2 9 18
3 2 6
3 3 9
3 4 12
3 9 27
4 2 8
4 3 12
4 4 16
4 9 36
9 2 18
9 3 27
9 4 36
9 9 81
outcomes have a product less than 20 and contain at least one even number are in bold letters.
Hence, outcomes have a product less than 20 and contain at least one even number are:
10
There are 8 outcomes of spinning the spinner twice that result in a product less than 20 and contain at least one even number.
To determine how many outcomes of spinning a spinner twice result in a product less than 20 and contain at least one even number, we start by listing all possible outcomes.
The spinner is divided into four equal sections: 2, 3, 4, and 9. When spun twice, there are 16 outcomes in total:
(2,2)(2,3)(2,4)(2,9)(3,2)(3,3)(3,4)(3,9)(4,2)(4,3)(4,4)(4,9)(9,2)(9,3)(9,4)(9,9)We now filter these outcomes to find those with a product less than 20 and at least one even number:
(2,2) - Product: 4(2,3) - Product: 6(2,4) - Product: 8(3,2) - Product: 6(3,4) - Product: 12(4,2) - Product: 8(4,3) - Product: 12(4,4) - Product: 16The total number of outcomes that meet the criteria is 8.
Please help on puzzle #2
Answer:
(2 +4i)(5 -6i) . . . or . . . (4 -2i)(6 +5i)
Step-by-step explanation:
The product of two complex numbers is ...
(a +bi)(c +di) = (ac -bd) +(bc +ad)i
So, we're looking for pairs of numbers that can be combined in different ways to give 34 and 8. The numbers we found (by trial and error) are ...
2, 4, 5, 6
where 4*6 +2*5 = 34 and 4*5 -2*6 = 8. Because of the effect if i^2 on the sign, we need to have the imaginary parts have opposite signs.
Each of the solutions shown above is representative of 4 solutions. For example, for the first one, you could have ...
(2 +4i)(5 -6i) = (2·5 +4·6) + (4·5 +2(-6))i = 34 +8i
(5 -6i)(2 +4i) = (5·2 +6·4) + (-6·2 +5·4) = 34 +8i . . . . . order of factors swapped
(-2 -4i)(-5 +6i) = ((-2)(-5) -(-4)(6)) + ((-4)(-5) +(-2)(6))i = 34 +8i . . . . both factors in the first solution negated
(-5 +6i)(-2 -4i) = ((-5)(-2) -(6)(-4)) +(6(-2) +(-5)(-4))i = 34 +8i . . . . factors swapped and negated
___
Likewise, the second shown solution above is representative of 4 solutions.
Possible solutions are ...
(2 +4i)(5 -6i)(4 -2i)(6 +5i)with sign and order variations.
_____
Comment on trial and error
Actually, we did an exhaustive search of the 441 products of single-digit numbers [-9, 9] to see which pairs of them differed by 34. Then, among those, we looked for product pairs that added to 8. In the end, we found the 8 solutions described above.
Which expression is equivalent to sec2xcot2x?
A.
sin2x
B.
csc2x
C.
`(1)/(cos^2x)`
D.
`(1)/(tan^2x)`
Answer:
Option B is correct answer.
Step-by-step explanation:
We need to solve the expression sec2xcot2x.
We know sec x = 1/ cos x and cot x = 1/ tan x and tan x = sin x/cos x and 1/tanx = cosx /sinx
Since in question we 2x instead of x so, replacing x with 2x and Putting values:
[tex]sec2x\,\, cot2x\\=\frac{1}{cos 2x} * \frac{1}{tan 2x} \\=\frac{1}{cos 2x} * \frac{cos2x}{sin2x}\\=\frac{1}{sin 2x}\\=csc2x[/tex]
So, Option B is correct answer.
Answer:
b
Step-by-step explanation:
In rhombus JKLM, if m<KLJ = 38°, find m<JIVIL.
Answer:
∠JML = 104°
Step-by-step explanation:
JL bisects angle KLM, so ...
∠KLM = 2·∠KLJ = 2·38° = 76°
Adjacent angles in any parallelogram are supplementary, so ...
∠JML = 180° -∠KLM = 180° -76°
∠JML = 104°
The expression is the result of applying the change of base formula to a logarithmic expression.
Which could be the original expression?
x= 8 is the correct answer
Answer:
x = 8
Step-by-step explanation:
Given equation is,
[tex]log_5(10x-1)=log_5(9x+7)[/tex]
We know that,
[tex]log_a(b)=log_a(c)\implies b = c[/tex]
[tex]\implies 10x -1 = 9x + 7[/tex]
Subtracting 9x on both sides,
[tex]x - 1 = 7[/tex]
Adding 1 on both sides,
[tex]x = 8[/tex]
Hence, the solution would be x = 8
A foam material has a density of 175 g/l. what is its density in units of lb/ft3? How do you get 1 gram/liter = 0.06242796 pound/cubic foot?
Answer:
Given: 175 g/L
1 gram/liter = 0.06242796 pound/cubic foot
175 g/L * 0.06242796 pound/cubic foot= 10.924893 lb/ft3
So, a foam material has a density of 10.924893 lb/ft3 in units of lb/ft3
Step-by-step explanation:
Can someone please help me with this problem
30 points!!!!
Answer:
29.4
Step-by-step explanation:
29.4444 rounded to the nearest tenth is 29.4
the other answerer forgot to round to the tenths place
Consider the following factor and indicate whether it increases or decreases the equilibrium price of gasoline and the equilibrium quantity of gasoline sold. In this problem, assume that gasoline is a normal good.
When the number of sellers decreases, the
Choose one:
A. supply curve shifts to the right.
B. demand curve shifts to the right.
C. demand curve shifts to the left.
D. supply curve shifts to the left.
As a result,
Choose one:
A. price decreases and quantity decreases.
B. price decreases and quantity increases.
C. price increases and quantity decreases.
D. price increases and quantity increases.
The answer for your question is A,D.
Answer:
D. supply curve shifts to the left
C. price increases and quantity decreases
Step-by-step explanation:
Since the number of sellers decreases, the quantity available at the same price decreases. This shifts the supply curve to the left.
When the supply curve shifts to the left, the equilibrium point shifts to the left (and up the demand curve). Hence the price increases and the quantity decreases.
1 pt) If a parametric surface given by r1(u,v)=f(u,v)i+g(u,v)j+h(u,v)k and −4≤u≤4,−4≤v≤4, has surface area equal to 1, what is the surface area of the parametric surface given by r2(u,v)=5r1(u,v) with −4≤u≤4,−4≤v≤4?
Surface area of [tex]\(r_2(u,v)\)[/tex] is [tex]\(25\)[/tex] times the area of parameter domain[tex]\(D\),[/tex] yielding [tex]\(1600\)[/tex] if [tex]\(D\)[/tex] is [tex]\(8 \times 8\).[/tex]
Let's break it down step by step:
step:-1. **Define the parametric surfaces**: We have two parametric surfaces: [tex]\( r_1(u,v) = f(u,v)i + g(u,v)j + h(u,v)k \)[/tex] and [tex]\( r_2(u,v) = 5r_1(u,v) \).[/tex]
step:-2. **Calculate the partial derivatives**: Compute the partial derivatives of [tex]\( r_1 \)[/tex] with respect to [tex]\( u \)[/tex] and [tex]\( v \)[/tex] denoted by [tex]\( r_{1u} \)[/tex] and [tex]\( r_{1v} \).[/tex]
step:-3. **Multiply by 5**: Since[tex]\( r_2(u,v) = 5r_1(u,v) \),[/tex] the partial derivatives of [tex]\( r_2 \)[/tex] with respect to [tex]\( u \)[/tex] and [tex]\( v \)[/tex] will be 5 times the corresponding partial derivatives of [tex]\( r_1 \)[/tex], denoted by [tex]\( r_{2u} \)[/tex] and [tex]\( r_{2v} \).[/tex]
step:-4. **Calculate the cross product**: Compute the cross product of [tex]\( r_{2u} \)[/tex] and[tex]\( r_{2v} \),[/tex] denoted by [tex]\( \| r_{2u} \times r_{2v} \| \)[/tex]. This will be 25 times the magnitude of the cross product of [tex]\( r_{1u} \)[/tex] and [tex]\( r_{1v} \),[/tex] as the cross product is linear with respect to the vectors involved.
step:-5. **Surface area integral**: Use the formula for the surface area integral: [tex]\( A = \iint_D \| r_u \times r_v \| \, dA \),[/tex] where [tex]\( \| r_{2u} \times r_{2v} \| \)[/tex] replaces [tex]\( \| r_{u} \times r_{v} \| \).[/tex]
step:-6. **Calculate the integral**: Integrate[tex]\( \| r_{2u} \times r_{2v} \| \)[/tex] over the parameter domain [tex]\( D \)[/tex]. Since [tex]\( \| r_{2u} \times r_{2v} \| \)[/tex] is constant and equal to 25 times the magnitude of the cross product of [tex]\( r_{1u} \)[/tex] and [tex]\( r_{1v} \)[/tex], the integral becomes [tex]\( 25 \times \text{Area of } D \).[/tex]
step:-7. **Determine the area of the parameter domain**: If the parameter domain [tex]\( D \)[/tex] is a rectangle with sides of length 8 in both directions, its area is [tex]\( 8 \times 8 = 64 \).[/tex]
step:-8. **Final calculation**: Multiply the area of [tex]\( D \)[/tex] by 25 to get the surface area of[tex]\( r_2(u,v) \)[/tex], which is[tex]\( 25 \times 64 = 1600 \).[/tex]
So, the surface area of the parametric surface given by [tex]\( r_2(u,v) = 5r_1(u,v) \)[/tex] is 1600.
The surface area of the parametric surface [tex]\( \mathf{r}_2(u, v) \)[/tex] is 25.
To find the surface area of the parametric surface given by [tex]\( \mathf{r}_2(u, v) = 5 \mathf{r}_1(u, v) \)[/tex] where [tex]\( -4 \leq u \leq 4 \)[/tex] and [tex]\( -4 \leq v \leq 4 \)[/tex], given that the surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex] over the same parameter range is 1, follow these steps:
Surface Area of [tex]\( \mathf{r}_1(u, v) \)[/tex]
The surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex] is given to be 1.
Relationship Between [tex]\( \mathf{r}_1 \) and \( \mathf{r}_2 \)[/tex]
[tex]\[ \mathf{r}_2(u, v) = 5 \mathf{r}_1(u, v) \][/tex]
Effect of Scaling on Surface Area
When a surface is scaled by a factor k, the surface area is scaled by a factor of k². This is because surface area is a two-dimensional measure, and scaling each dimension by k multiplies the area by k².
Calculation for [tex]\( \mathf{r}_2(u, v) \)[/tex]
In this problem, the scaling factor k is 5. Therefore, the surface area of [tex]\( \mathf{r}_2(u, v) \)[/tex] will be [tex]\( 5^2 \)[/tex] times the surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex].
[tex]\[\text{Surface area of } \mathf{r}_2(u, v) = 5^2 \times \text{Surface area of } \mathf{r}_1(u, v)\][/tex]
Substitute the Given Surface Area
The surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex] is 1.
[tex]\[\text{Surface area of } \mathf{r}_2(u, v) = 5^2 \times 1 = 25\][/tex]
Last answer choice is 4,2
Help
Use the midpoint formula:
(-7,-4) (-1,8)
(-7 + -1) /2 , (-4 +8) /2
-8/2 , 4/2
Midpoint = (-4,2)
A number is increased by 54. The sumos then divided by 9. The result is 21. Write an equation to represent the discription, use n for the number
Answer:
(n+54)/9 = 21
Step-by-step explanation:
A number: n
Is increased by 54: n+54
The sum is divided by 9: (n +54)/9
and the result is 21:
(n +54)/9 = 21
Find the value of x.
log 3 x=4
Answer:
The correct answer option is D. 81.
Step-by-step explanation:
We are given the following expression and we are to find the value of x:
[tex] log _ 3 x = 4 [/tex]
We can inverse [tex] log _ 3 x [/tex] and re-write it as [tex]3^x[/tex].
Doing that, we will raise both the sides of the equation to the power of 3 to get:
[tex] 3 ^ { log 3 x } = 3 ^ 4 [/tex]
x = 81
What is the average rate of change of the function over the interval x = 0 to x = 5?
f(x) = 2x^2 - 1
Enter your answer, as a fraction, in the box.
[tex]\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(x)= 2x^2-1\qquad \begin{cases} x_1=0\\ x_2=5 \end{cases}\implies \cfrac{f(5)-f(0)}{5-0} \\\\\\ \cfrac{[2(5)^2-1]~~-~~[2(0)^2-1]}{5}\implies \cfrac{50-(-1)}{5}\implies \cfrac{50+1}{5}\implies \cfrac{51}{5}\implies 10\frac{1}{5}[/tex]