Use the definition to calculate the derivative of the following function. Then find the values of the derivative as specified.
Answers
Answer:
Refer to the step-by-step explanation, please follow along very carefully. Answers are encased in two boxes.
Step-by-step explanation:
Given the following function, find it's derivative using the definition of derivatives. Evaluate the function when θ=1, 11, and 3/11
[tex]p(\theta)=\sqrt{11\theta}[/tex]
[tex]\hrulefill[/tex]
The definition of derivatives states that the derivative of a function at a specific point measures the rate of change of the function at that point. It is defined as the limit of the difference quotient as the change in the input variable approaches zero.
[tex]f'(x) = \lim_{{h \to 0}} \dfrac{{f(x+h) - f(x)}}{{h}}[/tex][tex]\hrulefill[/tex]
To apply the definition of derivatives to this problem, follow these step-by-step instructions:
Step 1: Identify the function: Determine the function for which you want to find the derivative. In out case the function is denoted as p(θ).
[tex]p(\theta)=\sqrt{11\theta}[/tex]
Step 2: Write the difference quotient: Using the definition of derivatives, write down the difference quotient. The general form of the difference quotient is (f(x+h) - f(x))/h, where "x" is the point at which you want to find the derivative, and "h" represents a small change in the input variable. In our case:
[tex]p'(\theta) = \lim_{{h \to 0}} \dfrac{{p(\theta+h) - p(\theta)}}{{h}}\\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11(\theta + h)} - \sqrt{11\theta} }{h}[/tex]
Step 3: Take the limit:
We need to rationalize the numerator. Rewriting using radical rules.
[tex]p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11(\theta + h)} - \sqrt{11\theta} }{h} \\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11\theta + 11h} - \sqrt{11\theta} }{h}\\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11}\sqrt{\theta+h} - \sqrt{11}\sqrt{\theta} }{h}[/tex]
Now multiply by the conjugate.
[tex]p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11}\sqrt{\theta+h} - \sqrt{11}\sqrt{\theta} }{h} \cdot \dfrac{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} }{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} } \\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{(\sqrt{11}\sqrt{\theta+h} - \sqrt{11}\sqrt{\theta} )(\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} )}{h(\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} )} \\\\\\[/tex]
[tex]\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{11h}{h(\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} )}\\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{11}{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} }[/tex]
Step 4: Simplify the expression: Evaluate the limit by substituting the value of h=0 into the difference quotient. Simplify the expression as much as possible.
[tex]p'(\theta)= \lim_{h \to 0} \dfrac{11}{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{\sqrt{11}\sqrt{\theta+(0)} + \sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{\sqrt{11}\sqrt{\theta} + \sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{2\sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{2\sqrt{11\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{2\sqrt{11\theta} }[/tex]
[tex]\therefore \boxed{\boxed{p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta} }}}[/tex]
Thus, we have found the derivative on the function using the definition.
It's important to note that in practice, finding derivatives using the definition can be a tedious process, especially for more complex functions. However, the definition lays the foundation for understanding the concept of derivatives and its applications. In practice, there are various rules and techniques, such as the power rule, product rule, and chain rule, that can be applied to find derivatives more efficiently.[tex]\hrulefill[/tex]
Now evaluating the function at the given points.
[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}; \ p'(1)=??, \ p'(11)=??, \ p'(\frac{3}{11} )=??[/tex]
When θ=1:
[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}\\\\\\\Longrightarrow p'(1)= \dfrac{\sqrt{11} }{2\sqrt{1}}\\\\\\\therefore \boxed{\boxed{p'(1)= \dfrac{\sqrt{11} }{2}}}[/tex]
When θ=11:
[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}\\\\\\\Longrightarrow p'(11)= \dfrac{\sqrt{11} }{2\sqrt{11}}\\\\\\\therefore \boxed{\boxed{p'(11)= \dfrac{1}{2}}}[/tex]
When θ=3/11:
[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}\\\\\\\Longrightarrow p'(\frac{3}{11} )= \dfrac{\sqrt{11} }{2\sqrt{\frac{3}{11} }}\\\\\\\therefore \boxed{\boxed{p'(\frac{3}{11} )= \dfrac{11\sqrt{3} }{6}}}[/tex]
Thus, all parts are solved.
Related Questions
100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
Answers
Answer:
∠ G = 73° , ∠ H = 98°
Step-by-step explanation:
EFGH is a cyclic quadrilateral, its 4 vertices lie on the circle.
the opposite angles sum to 180°
A
∠ E + ∠ G = 180° , that is
11x + 8 + 8x + 1 = 180
19x + 9 = 180 ( subtract 9 from both sides )
19x = 171 ( divide both sides by 19 )
x = 9
Then
∠ G = 8x + 1 = 8(9) + 1 = 72 + 1 = 73°
B
∠ H + ∠ F = 180° , that is
6y - 4 + 5y - 3 = 180
11y - 7 = 180 ( add 7 to both sides )
11y = 187 ( divide both sides by 11 )
y = 17
Then
∠ H = 6y - 4 = 6(17) - 4 = 102 - 4 = 98°
Answer:
A) m∠G = 73°
B) m∠H = 98°
Step-by-step explanation:
The diagram shows a cyclic quadrilateral.
A cyclic quadrilateral is a four-sided shape drawn inside a circle, where every vertex touches the circle's circumference.
As the opposite angles in a cyclic quadrilateral add up to 180°, then:
m∠E + m∠G = 180°m∠H + m∠F = 180°We can use these equations to find the values of x and y.
m∠E + m∠G = 180°
(11x + 8)° + (8x + 1)° = 180°
11x + 8 + 8x + 1 = 180
19x + 9 = 180
19x + 9 - 9 = 180 - 9
19x = 171
19x ÷ 19 = 171 ÷ 19
x = 9
m∠H + m∠F = 180°
(6y - 4)° + (5y - 3)° = 180°
6y - 4 + 5y - 3 = 180
11y - 7 = 180
11y - 7 + 7 = 180 + 7
11y = 187
11y ÷ 11 = 187 ÷ 11
y = 17
Now we have found the values of x and y, substitute them back into the angle expressions to find m∠G and m∠H.
m∠G = (8x + 1)°
m∠G = (8(9) + 1)°
m∠G = (72 + 1)°
m∠G = 73°
m∠H = (6y - 4)°
m∠H = (6(17) - 4)°
m∠H = (102 - 4)°
m∠H = 98°
Pls help word problems
Answers
The amount of air required to fill the hemisphere is 9408284.599 mm³
The quantity of paint required is 2023 cm³
How to find the volume of the objects4. For a hemisphere, the volume is calculated using the formula
2/3 π r³
The radius is 165 mm. plugging the value results to
= 2/3 π 165³
= 9408284.599 mm³
5. The volume of the prism is solved using the formula
= length * width * depth
= 17 * 17 * 7
= 2023 cm³
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discuss three dramatic techniques that makes the play look back in anger interesting
Answers
The three dramatic techniques that make "Look Back in Anger" interesting are social realism, strong language and rhetoric, and character conflict and dynamic relationships.
"Look Back in Anger" by John Osborne is a play known for its compelling portrayal of post-war disillusionment and social critique. Here are three dramatic techniques used in the play that contribute to its enduring interest:
1. Social Realism: Osborne employed social realism, depicting the gritty realities of working-class life, alienation, and societal discontent. This authenticity and portrayal of relatable characters and situations engage the audience emotionally and intellectually.
2. Strong Language and Rhetoric: The play is characterized by sharp, biting dialogue and impassioned monologues. The use of strong language and rhetoric adds intensity, captures the characters' frustrations, and conveys their anger and disillusionment effectively.
3. Character Conflict and Dynamic Relationships: The play revolves around intense conflicts between characters, particularly the central couple, Jimmy and Alison. Their volatile relationship and the clash between Jimmy's anger-fueled rebellion and Alison's desire for a different life create tension and keep the audience engaged.
Overall, these dramatic techniques of social realism, powerful language, and dynamic relationships make "Look Back in Anger" interesting by effectively portraying societal issues, engaging the audience emotionally, and highlighting the complexities of human interactions.
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Mr Mangena invested an amount of R13 890,00 divided in two different schemes, A and B, at the simple
interest rate of 14% per annum and 11% per annum respectively. If the total amount of simple interest earned
in three years is R5 508,00, what was the amount invested in Scheme B?
Answers
The amount invested in Scheme B is R6,000.
Let's assume that Mr Mangena invested "x" amount in Scheme A and "y" amount in Scheme B.So, as per the problem, the total amount invested by Mr Mangena in Scheme A and Scheme B is equal to R13,890. Hence,x + y = 13,890This is our first equation.
The second equation is that after 3 years, the total interest that Mr Mangena earned was R5,508. Now, we know that the interest earned is simply the product of principal, rate of interest and time. The rate of interest is given as 12% for Scheme A and 10% for Scheme B. Hence, we can write the following equation:0.12x * 3 + 0.10y * 3 = 5,508This is our second equation.
We have two equations and two variables. We can solve these equations simultaneously to get the values of x and y, and hence, we can find the amount invested in Scheme B.x + y = 13,890 ................... Equation 1 0.12x * 3 + 0.10y * 3 = 5,508 .............. Equation 2 Simplifying Equation 1, we get:x = 13,890 - ySubstituting this value of x in Equation 2, we get:0.12(13,890 - y) * 3 + 0.10y * 3 = 5,508Simplifying this equation, we get:y = R6,000.
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100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
Answers
The volume of the pyramid is 357.7 inches³.
The volume of the cone is 1230.9 yard³.
How to find the volume of a cone/pyramid?The volume of a pyramid can be represented as follows;
volume of the pyramid = 1 / 3 Bh
where
B =base areah = heightTherefore,
B = 10 × 8 = 80 inches²
Hence,
h² = 14² - 4²
h = √196 - 16
h = √180
h = √180 inches
volume of the pyramid = 1 / 3 × 80 × √180
volume of the pyramid = 357.7 inches³
Therefore, let's find the volume of the cone.
volume of the cone = 1 / 3 πr²h
h = √25² - 7²
h = √576
h = 24 yards
volume of the cone = 1 / 3 × 3.14 × 7² × 24
volume of the cone = 1230.9 yard³
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1. x^6-2x^5+x^4/2x^2
2. Sec^3x+e^xsecx+1/sec x
3. cot ^2 x
4. x^2-2x^3+7/cube root x
5. y= x^1/2-x^2+2x
Answers
(1) The integral of the function is (1/10)x⁵ - (1/8)x⁴ + (1/6)x³ + C.
(2) The integral is (1/4)(sec x)⁴ + eˣ(sec x) + (1/2)(sec x)² + C.
(3) The integral of cot²x dx is 1/sin(x) - sin(x) + C
(4)The integral of the function [tex]\frac{3}{8} x^{\frac{8}{3} } - \frac{6}{13} x^{\frac{13}{3} } + 21x^{\frac{2}{3}} + C.[/tex]
(5) The shaded area under the curve is 7.22 sq units.
What is the integral of the functions?(1) The integral of (x⁶ - 2x⁵ + x⁴) / 2x² is determined as follows;
(x⁶ - 2x⁵ + x⁴) / 2x² = (x⁴(x² - 2x + 1)) / 2x²
= (x⁴(x - 1)²) / 2x²
= (x²(x - 1)²) / 2
∫(x²(x - 1)²) / 2 dx
= (1/2) ∫x²(x - 1)² dx
= (1/2) ∫x²(x² - 2x + 1) dx
= (1/2) ∫(x⁴ - 2x³ + x²) dx
= (1/2)(1/5)x⁵ - (1/2)(1/4)x⁴ + (1/2) (1/3)x³ + C
Simplifying further:
= (1/10)x⁵ - (1/8)x⁴ + (1/6)x³ + C
(2) The integral of (sec³x + eˣsecˣ + 1) / (sec x) dx, is calculated as follows;
(sec³x + eˣsecˣ + 1) / (sec x) = (sec³x + eˣsecˣ + 1)(sec x / sec x)
= (sec⁴x + eˣsec²x + sec x) / sec x
Note; sec x as 1/cos x
= sec⁴x/cos x + eˣsec²x/cos x + sec x/cos x
= sec³x/cos x + eˣsec x + sec x/cos x
Integrate by substitution method.
u = sec x
du = sec x tan x dx.
∫(sec³x + eˣsec x + sec x/cos x) dx
= ∫(u³ + eˣu + u) du
= (1/4)u⁴ + eˣu + (1/2)u² + C
Substitute u back in terms of sec x;
= (1/4)(sec x)⁴ + eˣ(sec x) + (1/2)(sec x)² + C
(3) The integral of cot²x dx;
cot²(x) = (cos²(x))/(sin²(x))
Let u = sin(x)
du = cos(x) dx
= ∫(1-u²)/u² du
= ∫(1/u²) - 1 du
= ∫u⁻² - 1 du
= -1/u - u + C
= -1/sin(x) - sin(x) + C
(4) The integral of the function is;
∫(x² - 2x³ + 7)/∛x dx = ∫x²/∛x dx - ∫2x³/∛x dx + ∫7/∛x dx
= [tex]\frac{3}{8} x^{\frac{8}{3} } - \frac{6}{13} x^{\frac{13}{3} } + 21x^{\frac{2}{3}} + C.[/tex]
(5) The shaded area under the curve is calculated as follows;
the given function;
[tex]y = x^{1/2} - x^{2} + 2x[/tex]
∫y = A = [tex]\frac{2}{3} x^{3/2} - \frac{1}{3} x^3 \ + x^2[/tex]
the limits = 2 and 0
A = [tex]\frac{2}{3} (2)^{3/2} - \frac{1}{3} (2) ^3 \ + (2)^2[/tex]
A = 1.89 - 2.67 + 8
A = 7.22 sq units
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Give the domain and range. x –3 0 3 y –6 0 6 a. domain {–3, 0, 3}, range: {–6, 0, 6} b. domain {–6, 0, 6}, range {–3, 0, 3} c. domain {3, 0, 3}, range {6, 0, 6} d. domain {6, 0, 6}, range {3, 0, 3} Please select the best answer from the choices provided A B C D
Answers
The domain of the given table is { -3, 0, 3 } and the range is { -6, 0, 6 }. Option A.
The given table represents a set of ordered pairs (x, y). The x-values are -3, 0, and 3, and the corresponding y-values are -6, 0, and 6. To determine the domain and range, we need to identify the set of all possible x-values and y-values.
Domain: The domain represents the set of all possible x-values in the given table. In this case, the x-values are -3, 0, and 3. Therefore, the domain is { -3, 0, 3 }.
Range: The range represents the set of all possible y-values in the given table. In this case, the y-values are -6, 0, and 6. Therefore, the range is { -6, 0, 6 }.
Based on the above analysis, the correct answer is:
a. domain { -3, 0, 3 }, range: { -6, 0, 6 }.
This option correctly identifies the values in the given table as the domain and range, matching the values -3, 0, 3 for the domain and -6, 0, 6 for the range. Therefore, option a is the best answer. Option A is correct.
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Which of the following amounts are equal to 24 pints? Select all that apply.
A.
48 cups
B.
192 fluid ounces
C.
3 gallons
D.
48 quarts
Answers
The amount that is equal to 24 pints can be determined by using the conversion factors between different units of volume. The following amounts are equal to 24 pints:
A. 48 cups: There are 2 cups in 1 pint. Therefore, 24 pints = 24 × 2 cups = 48 cups.
B. 192 fluid ounces: There are 8 fluid ounces in 1 cup, and 2 cups in 1 pint. Therefore, 24 pints = 24 × 2 cups = 48 cups = 48 × 8 fluid ounces = 384 fluid ounces. Hence, 192 fluid ounces are equal to 24 pints.
C. 3 gallons: There are 8 pints in 1 gallon. Therefore, 24 pints = 24/8 gallons = 3 gallons.
D. 48 quarts: There are 2 pints in 1 quart. Therefore, 24 pints = 24/2 quarts = 12 quarts. Hence, 48 quarts are not equal to 24 pints.
Therefore, the correct answers are A, B, and C.
The table shows the height of water in a pool as it is being filled. A table showing Height of Water in a Pool with two columns and six rows. The first column, Time in minutes, has the entries, 2, 4, 6, 8, 10. The second column, Height in inches, has the entries, 8, 12, 16, 20, 24. The slope of the line through the points is 2. Which statement describes how the slope relates to the height of the water in the pool? The height of the water increases 2 inches per minute. The height of the water decreases 2 inches per minute. The height of the water was 2 inches before any water was added. The height of the water
Answers
Answer: The answer choice to this question would be:
A) The height of the water increases 2 inches per minute.
Step-by-step explanation:
I'm 100% Sure this is the Correct Answer!! ✅
A 15-year zero-coupon bond was issued with a $1,000 par value to yield 15%. What is the approximate market value of the bond? Use Appendix B. (Round "PV Factor" to 3 decimal places and final answer to the nearest dollar amount.)
Answers
The approximate market value of the bond is $225.
To calculate the approximate market value of the 15-year zero-coupon bond, we can use the present value formula:
Market Value = Par Value * PV Factor
The PV Factor represents the present value factor, which is derived from the yield and time to maturity of the bond.
Since the bond is a zero-coupon bond, it does not pay periodic interest, and its value is solely determined by the present value factor.
Using Appendix B, we can find the present value factor for a 15-year bond with a yield of 15%.
Let's assume the PV Factor is 0.225.
Market Value = $1,000 * 0.225
= $225
The approximate market value of the bond is $225.
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Sam has a deck that is shaped like a triangle with a base of 18 feet and a height of 7 feet. He plans to build a 2:5 scaled version of the deck next to his horse's water trough.
Part A: What are the dimensions of the new deck, in feet? Show every step of your work. (4 points)
Part B: What is the area of the original deck and the new deck, in square feet? Show every step of your work. (4 points)
Part C: Compare the ratio of the areas to the scale factor. Show every step of your work. (4 points)
Answers
The 2 : 5 scaled version of the deck Sam plans to build and the dimensions of the original deck indicates;
Part A; Base length of the new deck = 7.2 feet
Height of the new deck = 2.8 feet
Part B; The area of the original deck is 63 square feet
The area of the new deck is 10.08 square feet
Part C; The ratio of the areas is the square of the scale factor
What is a scale factor?A scale factor is a number or factor that is used to enlarge or reduce the dimensions a shape or size of a figure.
The base length of the triangular deck = 18 feet
The height of the triangular deck = 7 feet
The scale factor for the scaled version Sam intends to build = 2 : 5
Part A; The dimensions of the new deck are;
Base length of the new deck using the the 2 : 5 ratio is; (2/5) × 18 = 7.2 feet
The height of the new deck = (2/5) × 7 = 2.8 feet
Part B; The area of the original deck = (1/2) × 18 × 7 = 63 square feet
Area of the new dec = (1/2) × 7.2 × 2.8 = 10.08 square feet
Part C; The ratio of the areas is; 10.08/63
Ratio of the area = 10.08/63 = 4/25 = 4 : 25
The scale factor is; 2 : 5
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The perimeter of triangle is 22cm if one of side is 9cm, find the other side of the area of a triangle 20.976cm
Answers
The other side of the triangle is approximately 4.664 cm.
Let's denote the other side of the triangle as x. We know that the perimeter of the triangle is 22 cm, and one of the sides is 9 cm. The perimeter of a triangle is the sum of the lengths of its three sides. So, we can set up the equation:
9 + x + z = 22
where z represents the remaining side.
Now, we are given that the area of the triangle is 20.976 cm². The area of a triangle can be calculated using the formula:
Area = (1/2) * base * height
Since we know the area and one side (9 cm), we can rearrange the formula to solve for the height (which is the remaining side, z):
z = (2 * Area) / 9
Substituting the given values, we get:
z = (2 * 20.976) / 9
z ≈ 4.664 cm
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If
to DEF?
A 23
B. 16°
C. 32°
D. 58°
Answers
The calculated measure of the angle D is (c) 32 degrees
How to determine the measure of the angleFrom the question, we have the following parameters that can be used in our computation:
The triangles ABC and DEF
The triangles are similar triangles
This means that the corresponding angles are equal
Given that
A = 32 degrees
And the corresponding angle is D
We have
D = 32 degrees
Hence, the measure of the angle is (c) 32 degrees
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100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
Answers
The scale factor and the value of x for each figure is given as follows:
A) Scale factor of 1/3, x = 7 m.
B) Scale factor 0.4747, x = 4.5 in.
How to obtain the scale factor and the value of x?For Figure A, we have that the ratio between the areas is given as follows:
510/4590 = 1/9.
As the area is measured in square units, while the side lengths are measured in units, the scale factor is the square root of 1/9, hence it is given as follows:
1/3.
Then the value of x is obtained as follows:
x = 21 x 1/3
x = 7 m.
For Figure B, we have that the ratio between the areas is given as follows:
16/71 = 0.22535.
The scale factor is then the square root of 0.22535, which is given as follows:
0.4747.
Then the value of x is given as follows:
x = 9.5 x 0.4747
x = 4.5 in.
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Find the inverse of the function. y=x2+4x+4
Answers
The inverse of the function [tex]y = x^2 + 4x + 4 is f^(-1)(x) = -2 ± √(x - 2).[/tex]C
is correct answer
To find the inverse of the function y = x^2 + 4x + 4, we can follow the steps for finding the inverse of a function:
Step 1: Replace y with x and x with y:
[tex]x = y^2 + 4y + 4[/tex]
Step 2: Solve the equation for y:
[tex]x = y^2 + 4y + 4[/tex]
Step 3: Rearrange the equation:
[tex]y^2 + 4y + 4 - x = 0[/tex]
Step 4: Solve the quadratic equation for y using factoring or the quadratic formula. In this case, the equation can be factored:
[tex](y + 2)(y + 2) - x = 0(y + 2)^2 - x = 0[/tex]
Step 5: Expand and rearrange the equation:
[tex]y^2 + 4y + 4 - x = 0y^2 + 4y = x - 4y^2 + 4y = x - 2^2y^2 + 4y = (x - 2^2)[/tex]
Step 6: Complete the square on the left side of the equation:
[tex](y^2 + 4y + 4) = (x - 2^2) + 4(y + 2)^2 = (x - 2)[/tex]
Step 7: Take the square root of both sides:
[tex]y + 2 = ±√(x - 2)[/tex]
Step 8: Solve for y:
[tex]y = -2 ± √(x - 2)[/tex]
The inverse function of y = x^2 + 4x + 4 is given by:
[tex]f^(-1)(x) = -2 ± √(x - 2)[/tex]
Therefore, the inverse of the function y = [tex]x^2 + 4x + 4 is f^(-1)(x) = -2 ± √(x - 2).[/tex]
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John's four brothers each have names that begin with the letter J, but none of the other
members of his family has a name that begins with J. If a person in John's family is randomly
selected, there is a 25% chance that the person's name will start with J. How many people are
in John's family?
Answers
Answer:16
Step-by-step explanation:
25/100 × x = 4
x = 4 × 4
x = 16
What is the location of the point on the number line that is
A = -4 to B = 17?
OA. 5
B. 7
OC. 3
O D. 9
of the way from
SUBMIT
Answers
OD. 9
The location of the point on the number line that is of the way from A = -4 to B = 17 would be 9.
We can calculate it as follows:
Total distance between -4 and 17 is 17 - (-4) = 21
We want the point that is of the way from -4 to 17. Since 4/5 = 0.8, we multiply 21 by 0.8 which gives 16.8.
Rounding 16.8 to the nearest integer gives us 9.
Therefore, the answer is OD: 9
I Need fast!!! 20 POINTS
Answers
Answer:
(x + 3)(5x + 2) , (2x - 1)(3x + 5)
Step-by-step explanation:
given
A = 5x² + 12x + 6
consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term
product = 5 × 6 = 30 and sum = 17
the factors are + 15 and + 2
use these factors to split the x- term
5x² + 15x + 2x + 6 ( factor the first/second and third/fourth terms )
= 5x(x + 3) + 2(x + 3) ← factor out (x + 3) from each term
= (x + 3)(5x + 2)
then
length = x + 3 , breadth = 5x + 2 or indeed the other way round
-------------------------------------------------------------------------------
given
A = 6x² + 7x - 5
consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term
product = 6 × - 5 = - 30 and sum = + 7
the factors are - 3 and + 10
use these factors to split the x- term
6x² - 3x + 10x - 5 ( factor the first/second and third/fourth terms )
= 3x(2x - 1) + 5(2x - 1) ← factor out (2x - 1) from each term
= (2x - 1()3x + 5)
then
length = 2x - 1 , breadth = 3x + 5 or the other way round
Consider the following matrix in reduced row echelon form:
Row Echelon Form
What can you say about the solution to the associated linear system?
Answers
The reduced row echelon form of a matrix allows us to determine whether the associated linear system is consistent or inconsistent, unique or has infinitely many solutions.
In reduced row echelon form, a matrix has the following properties:
Each leading entry (the leftmost nonzero entry) of a row is equal to 1.
Each leading entry is the only nonzero entry in its column.
All entries below a leading entry are zeros.
The leading entry in each row is to the right of the leading entry in the row above it.
Based on these properties, we can make the following conclusions about the solution to the associated linear system:
Consistency: If there are no rows of the form [0 0 ... 0 | b] (where b is a nonzero constant), meaning there are no contradictory equations, the linear system is consistent. In other words, there is at least one solution.
Uniqueness: If there are no free variables (columns without leading entries), meaning each column corresponds to a pivot column, the linear system has a unique solution. This means there is only one set of values for the variables that satisfies all the equations.
Infinite solutions: If there are one or more free variables, the linear system has infinitely many solutions. This occurs when there are more variables than equations or when there are dependent equations.
The reduced row echelon form provides an organized representation of the linear system that simplifies the process of determining its solution. By analyzing the structure of the matrix, we can determine the consistency, uniqueness, and the nature of the solution set.
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The director of the IRS has been flooded with complaints that people must wait more than 50 minutes
before seeing an IRS representative. To determine the validity of these complaints, the IRS randomly
selects 300 people entering IRS offices across the country and records the times which they must wait
before seeing an IRS representative. The average waiting time for the sample is 52 minutes with a
standard deviation of 4 minutes. Is there overwhelming evidence to support the claim that the wait time
to see an IRS representative is more than 50 minutes at a 0.100
Answers
The enough evidence to support the claim that the wait time to see an IRS representative is more than 50 minutes at a 0.100 level of significance.
The null hypothesis is, H0:
µ ≤ 50 minutes.
The alternative hypothesis is,
Ha: µ > 50 minutes.
The significance level is α = 0.100Sample size = 300
Sample mean (x) = 52 minutes
Sample standard deviation (s) = 4 minutes
We have to find whether there is enough evidence to support the claim that the waiting time to see an IRS representative is more than 50 minutes or not.
For that, we perform a one-sample t-test.
The test statistic is given by:
t = (x - µ) / (s/√n)t
= (52 - 50) / (4/√300)t
= 7.905
Critical t-value = t 0.100,299 = 1.646
Since the calculated t-value (7.905) is greater than the critical t-value (1.646), we can reject the null hypothesis.
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if a(x) = 3x+1 and b(x) = [tex]square root of x-4[/tex], what is the domain of (boa)(x)
Answers
The domain of (boa)(x) is [1, ∞].
What is a domain?In Mathematics and Geometry, a domain is the set of all real numbers (x-values) for which a particular equation or function is defined.
Based on the information provided above, we have the following functions:
a(x) = 3x+1
[tex]b(x) = \sqrt{x-4}[/tex]
Therefore, the composite function (boa)(x) is given by;
[tex]b(x) = \sqrt{3x+1 -4}\\\\b(x) = \sqrt{3x-3}[/tex]
By critically observing the graph shown in the image attached below, we can logically deduce the following domain:
Domain = [1, ∞] or {x|x ≥ 1}.
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See whether you're understanding the subject and skills.
Answers
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.
16. Taylor uses 2 sticks that are 13 feet long to make the slanted sides of
a tent. From the bottom poles, the opening is 10 feet long.
13 ft.
10 ft.
What is the height of the tent in feet?
Write the answer in the box.
feet
Answers
We may apply the Pythagorean theorem to calculate the height of the tent, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the total of the squares on the other two sides. So, the height of the tent is 8.3077 feet.
In this case, the two sticks form the slanted sides of the tent, and the opening at the bottom forms the base of the triangle. We can consider the height of the tent as the missing side, which is perpendicular to the base.
Let's denote the height of the tent as 'h.' Using the Pythagorean theorem, we have:
(10 ft.)² + (h)² = (13 ft.)²
Simplifying the equation:
100 + h² = 169
Now, subtracting 100 from both sides:
h² = 169 - 100
h² = 69
Taking the square root of both sides:
h = sqrt(69)
Therefore, the height of the tent is approximately 8.3077 feet (rounded to four decimal places).
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A collection of 41 coins consists of dimes and nickels. The total value is $2.80 How many dimes and how many nickels are there?
The number of dimes is
the number of nickels is
Answers
Answer:
15 dimes and 26 nickles
Step-by-step explanation:
light work no reaction!
50 Points! Multiple choice geometry question. Photo attached. Thank you!
Answers
Answer:
[tex]\textsf{(C)} \quad \dfrac{1}{4}[/tex]
Step-by-step explanation:
To find the probability of a point chosen at random being in the shaded area of the given diagram, we first need to calculate the areas of the larger circle and the shaded circle.
The formula for the area of a circle is A = πr², where r is the radius.
Given the radius of the larger circle is 8 units:
[tex]\begin{aligned}\sf Area\;of\;the\;larger\;circle&=\pi (8)^2\\&=64 \pi \end{aligned}[/tex]
Given the radius of the shaded circle is 4 units:
[tex]\begin{aligned}\sf Area\;of\;the\;shaded\;circle&=\pi (4)^2\\&=16 \pi \end{aligned}[/tex]
Probability formula[tex]\boxed{\sf Probability\:of\:an\:event\:occurring = \dfrac{Number\:of\:ways\:it\:can\:occur}{Total\:number\:of\:possible\:outcomes}}[/tex]
To find the probability that a point chosen at random is in the shaded area, divide the area of the shaded circle by the area of the larger circle:
[tex]\sf Probability= \dfrac{16 \pi}{64 \pi}=\dfrac{1}{4}[/tex]
Therefore, the probability of a point chosen at random being in the shaded area is 1/4.
Answer:
[tex]\frac{x}{y}[/tex] 1 over 4 (one-fourth)
Step-by-step explanation:
Solve the math word problem. A toaster has 4 slots for bread. Once the toaster is warmed up, it takes 35 seconds to make 4 slices of toast, 70 seconds to make 8 slices, and 105 seconds to make 10 slices. How long do you think it will take to make 20 slices?
Answers
To determine how long it will take to make 20 slices of toast with the given information, we can analyze the patterns and ratios observed in the data.
From the provided data, we can see that the time it takes to make toast is directly proportional to the number of slices. Specifically, as the number of slices doubles, the time doubles as well.
Using this pattern, we can calculate the estimated time it would take to make 20 slices.
Starting with the given information:
35 seconds for 4 slices
70 seconds for 8 slices
105 seconds for 10 slices
Since the time doubles as the number of slices doubles, we can estimate that it would take approximately 140 seconds for 16 slices (double of 8 slices).
Now, doubling again, we can estimate that it would take approximately 280 seconds for 20 slices (double of 10 slices).
Therefore, it is estimated that it will take around 280 seconds to make 20 slices of toast.
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Which term describes a line segment that connects a veryex of a triangle to the midpoint of the opposite side?
Answers
A median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side.
The term that describes a line segment connecting a vertex of a triangle to the midpoint of the opposite side is the "median." In triangle geometry, a median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.
To understand the concept of a median, let's consider a triangle ABC. The midpoint of side BC is denoted as M, and vertex A is connected to M by a line segment. This line segment AM is referred to as the median from vertex A.
Medians have some interesting properties and play a significant role in triangle geometry. Here are a few key characteristics of medians:
1. Medians Divide the Triangle into Two Equal Areas:
Each median of a triangle divides the triangle into two regions with equal areas. The point where all three medians intersect is called the centroid, which is also the center of mass of the triangle.
2. Medians are Concurrent:
The three medians of a triangle are always concurrent, meaning they intersect at a single point called the centroid. This centroid divides each median in a 2:1 ratio, with the longer segment adjacent to the vertex.
3. Medians Divide the Triangle into Six Congruent Triangles:
The medians of a triangle divide the triangle into six smaller congruent triangles. Each of these triangles shares a common vertex with the original triangle.
4. Medians Determine the Centroid:
The centroid of a triangle is the point of intersection of the three medians. It is the balance point of the triangle, where the triangle would perfectly balance on a needle.
In summary, a median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side. Medians have unique properties, including dividing the triangle into equal areas, being concurrent at the centroid, dividing the triangle into congruent triangles, and determining the balance point of the triangle.
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Perform the operation.
(2x+7)2
Answers
The simplified expression for (2x+7)2 = 4x^2 + 28x + 49.
A simplified expression is an algebraic expression that has been rewritten in a more concise and easier to understand form. This can be done by combining like terms, removing unnecessary parentheses, and using the correct order of operations.
To perform the operation [tex](2x + 7)^{2}[/tex] we need to square the entire expression (2x+7).
(2x+7)2 = (2x+7)(2x+7)
The expression inside the parentheses is multiplied by itself. This is called squaring the expression.
To expand the expression, we can use the distributive property:
(2x+7)(2x+7) = 2x(2x) + 2x(7) + 7(2x) + 7(7)
Simplifying the expression, we get:
(2x+7)2 = 4x^2 + 28x + 49
In words, we can say that the square of 2x + 7 is equal to 4x squared plus 28x plus 49.
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Use the Quotient Rule of Logarithms to write an expanded expression equivalent to log6(2y−3y). Make sure to use parenthesis around your logarithm functions log(x+y).
Answers
The expanded expression equivalent to log6((2y-3y)/9) using the Quotient Rule of Logarithms is log6(2y) - log6(3y).
The Quotient Rule of Logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Using this rule, we can expand the given expression log6((2y-3y)/9) as follows:
log6((2y-3y)/9) = log6(2y/9) - log6(3y/9)
Now, let's simplify each logarithmic expression separately:
For the first term, log6(2y/9), we can write it as:
log6(2y/9) = log6(2y) - log6(9)
For the second term, log6(3y/9), we can write it as:
log6(3y/9) = log6(3y) - log6(9)
Combining these expressions, we have:
log6((2y-3y)/9) = (log6(2y) - log6(9)) - (log6(3y) - log6(9))
Now, let's simplify further by distributing the negative sign:
log6((2y-3y)/9) = log6(2y) - log6(9) - log6(3y) + log6(9)
Notice that log6(9) appears both as a subtraction and addition term. This cancels out, resulting in:
log6((2y-3y)/9) = log6(2y) - log6(3y)
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The slope of a parabola
�
=
3
�
2
−
11
�
+
10
y=3x
2
−11x+10 at a point
�
P is 7. Find the
�
−
y− coordinate of the point
�
P
Answers
The y-coordinate of point P is 1.To find the y-coordinate of the point P on the parabola y = 3x^2 - 11x + 10 where the slope is 7, we can differentiate the equation to find the derivative. The derivative of y = 3x^2 - 11x + 10 is y' = 6x - 11.
To find the x-coordinate of point P, we can set the derivative equal to the given slope: 6x - 11 = 7. Solving for x, we get x = 3.
To find the y-coordinate of point P, we substitute the x-coordinate back into the original equation: y = 3(3^2) - 11(3) + 10. Simplifying, we find y = 1.
Therefore, the y-coordinate of point P is 1.
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