Answer:
Here is how to calculate 716 divided by 4 using the long division method:
179
----------
4 | 716
4
---
31
28
---
8
Therefore, 716 divided by 4 is equal to 179.
Solve. 5x^2−35=0 The solution(s) is/are x= (Simplify your answer. Type an exact answer, using radicals as needed. Express complex numbers in terms of i. Use a comma to separate answers as needed.)
The solutions to the equation 5x^2 - 35 = 0 are x = ±√7.
To solve this quadratic equation, we can first isolate the variable by moving the constant term to the other side:
5x^2 = 35
Next, we divide both sides of the equation by 5 to solve for x^2:
x^2 = 7
To find the value of x, we take the square root of both sides:
√(x^2) = ±√7
Since we took the square root, we need to consider both the positive and negative square roots, giving us two solutions:
x = ±√7
Therefore, the solutions to the equation 5x^2 - 35 = 0 are x = ±√7.
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Given the sequence a). Find the next 3 terms. b). Find a₅₂ and a₇₅₈ (i.e. the 52nd and 758th terms). Show your work
a) The next 3 terms are aₙ + 3, aₙ + 6, and aₙ + 9, where aₙ is the last term in the sequence.
b) a₅₂ = 154 and a₇₅₈ = 2,272, using the formula aₙ = a₁ + (n - 1)d, with a₁ = 1 and d = 3.
a) To find the next 3 terms in the given sequence, we observe that each term is obtained by adding 3 to the previous term. Therefore, we can continue the pattern by adding 3 to the last term. In general, if the last term is denoted as aₙ, then the next three terms would be aₙ + 3, aₙ + 6, and aₙ + 9.
b) To find a specific term in the sequence, we can use the formula aₙ = a₁ + (n - 1)d, where a₁ is the first term, n is the term number, and d is the common difference. By substituting the given values (a₁ = 1 and d = 3) into the formula, we can find a₅₂ and a₇₅₈. Applying the formula, we find that a₅₂ = 154 and a₇₅₈ = 2,272.
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017) A student pilot filed a flight plan which included flying due west from an airport in Dallas,
Texas for 100 miles, then turning due north and flying 75 miles to land at an airport in
Wichita Falls, Texas. How far would he then have to fly in a straight line distance to get
back to Dallas?
The student pilot would have to fly approximately 125 miles in a straight-line distance to get back to Dallas.
How to determine the straight-line distance the student pilot would have to fly to get back to DallasThe distance flown due west from Dallas is 100 miles, and the distance flown due north from Wichita Falls is 75 miles. These distances form the two sides of a right-angled triangle.
Using the Pythagorean theorem, we can calculate the hypotenuse (the straight-line distance) as follows:
Hypotenuse² = (Distance due west)² + (Distance due north)²
Hypotenuse² = 100² + 75²
Hypotenuse² = 10000 + 5625
Hypotenuse² = 15625
Taking the square root of both sides gives us:
Hypotenuse = √15625
Hypotenuse ≈ 125
Therefore, the student pilot would have to fly approximately 125 miles in a straight-line distance to get back to Dallas.
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If p(x) is the image of y=3x^(2)+30x+2 after a translation right 4 units and up 5 units, write the equation of p(x) in the standard form of a quadratic function and describe its graph
The equation of p(x) in the standard form of a quadratic function after the translation right 4 units and up 5 units is p(x) = 3x² + 6x - y + 65.
The quadratic function is y = 3x² + 30x + 2. To translate it right 4 units, we substitute x with (x - 4). To translate it up 5 units, we substitute y with (y + 5).
So the new equation becomes y + 5 = 3(x - 4)² + 30(x - 4) + 2.
Expanding and simplifying, we get y + 5 = 3x² + 6x - y + 65.
Rearranging the terms, we obtain p(x) = 3x² + 6x - y + 65.
The graph of the quadratic function p(x) will have the same shape as y = 3x², but it will be shifted 4 units to the right and 5 units up compared to the original function. The vertex of the graph will be at the point (-2, 5), and the parabola will open upward.
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Find the slope of each line whose equation is given. Then, determine whether the lines are parallel, perpendicular, or neither. 1) y=6x−2 and y=6x+7 2) y=2x+4 and x+2y+10=0 3) y=8x−1 and 7x−y−1=0
1) The slope of y = 6x − 2 and y = 6x + 7 are both 6, thus, making them parallel.
2) The slope of y = 2x + 4 and x + 2y + 10 = 0 is 2 and -1/2, respectively. They are perpendicular.
3) The slope of y = 8x − 1 is 8 and of 7x − y − 1 = 0 is 7 and they are neither parallel nor perpendicular with each other.
1. To find the slope of the given lines, we must write their equations in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
y = 6x - 2 has a slope of 6
y = 6x + 7 has a slope of 6.
The two equations have the same slopes but different y-intercept. Therefore, the two lines are parallel.
2. y = 2x + 4 has a slope of 2.
Rearranging x + 2y + 10 = 0 into slope-intercept form gives 2y = -x - 10, so y = (-1/2)x - 5, which has a slope of -1/2.
-1/2 is the negative reciprocal of 2. Therefore, the two lines are perpendicular.
3. y = 8x - 1 has a slope of 8.
Rearranging 7x - y - 1 = 0 into slope-intercept form gives y = 7x - 1, which has a slope of 7.
The slopes of the two equation are neither the same nor the negative reciprocal of one another. Therefore, the two lines are neither parallel nor perpendicular.
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Determine the value of y for the inequality 3 times the quantity y plus one fourth end quantity is less than or equal to three fourths.
Answer:any number less than or equal to 0.
Step-by-step explanation:
To determine the value of y for the given inequality, let's solve it step by step:
3(y + 1/4) ≤ 3/4
First, let's simplify the left side of the inequality:
3y + 3/4 ≤ 3/4
Next, let's isolate the term with y by subtracting 3/4 from both sides of the inequality:
3y ≤ 3/4 - 3/4
This simplifies to:
3y ≤ 0
To solve for y, divide both sides of the inequality by 3:
y ≤ 0/3
y ≤ 0
Therefore, the value of y that satisfies the inequality is any number less than or equal to 0.
Answer:
number less than or equal to 0.
Step-by-step explanation:
Please help me with this
Find an equation of the circle with center \( (1,2) \) that passes through \( (-3,3) \)
The equation of the circle is:(x - 1)² + (y - 2)² = 17
To find the equation of a circle with center at (1, 2) and passing through (-3, 3), we need to use the formula for the standard form of the equation of a circle.
A circle with center (h, k) and radius r is given by the equation:(x - h)² + (y - k)² = r²
Substituting the given values, we have:(x - 1)² + (y - 2)² = r²
We can now find the value of r using the fact that the circle passes through the point (-3, 3):
(x - 1)² + (y - 2)² = r²(-3 - 1)² + (3 - 2)² = r²16 + 1 = r²17 = r²
So the equation of the circle is:(x - 1)² + (y - 2)² = 17
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Look at this rectangle.
4 1/4 cm
The perimeter of this rectangle is 13 1/2
What is the width?
The width of the rectangle is 5/2 cm, which can be written as 2 1/2 cm or 2.5 cm.
Let's denote the width of the rectangle as "w" (in cm). We are given that the length of the rectangle is 4 1/4 cm.
The perimeter of a rectangle is calculated by adding up the lengths of all four sides. In this case, we are given that the perimeter is 13 1/2 cm.
The formula for the perimeter of a rectangle is:
Perimeter = 2(length + width)
Substituting the given values:
13 1/2 = 2(4 1/4 + w)
To simplify the equation, we convert all mixed numbers to improper fractions:
Perimeter = 2(17/4 + w)
Next, we distribute the 2:
13 1/2 = 34/4 + 2w
Now, let's simplify the equation by finding a common denominator:
13 1/2 = 8 1/2 + 2w
We subtract 8 1/2 from both sides to isolate 2w:
5 = 2w
Finally, we divide both sides by 2 to solve for w:
w = 5/2
Therefore, the width of the rectangle is 5/2 cm, which can be written as 2 1/2 cm or 2.5 cm.
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The graphs of \( f \) and \( g \) are given. Find a formula for the function \( g \). \[ g(x)=x^{2}-9 x+9 \]
The formula for the function g is g(x) = x^2 - 9x + 9.
To find a formula for the function g, we can analyze the given equation. The equation g(x) = x^2 - 9x + 9 represents a quadratic function.
The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.
Comparing the given equation to the general form, we can determine the values of a, b, and c.
From the given equation, we have a = 1, b = -9, and c = 9. Therefore, the formula for the function g is:
g(x) = x^2 - 9x + 9
So, the formula for the function g is g(x) = x^2 - 9x + 9.
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Question-
The graphs of ( f ) and ( g ) are given. Find a formula for the function ( g )
equation lnA=lnA
0
−kt Where A
0
is the original amount of the substance, A is the amount of the substance remaining after time t, and k is a constant that is characteristic of the substance. For the radioactive isotope lead-214, k is 2.59×10
−2
minutes
−1
. If the original amount of lead-214 in a sample is 51.3mg, how much lead-214 remains after 31.6 minutes have passed? m9
After 31.6 minutes have passed, approximately 40.3 mg of lead-214 remains in the sample. This can be determined using the decay equation lnA = lnA₀ - kt, where A represents the amount of the substance remaining after time t, A₀ is the original amount of the substance, k is a constant characteristic of the substance, and t is the elapsed time.
The given equation, lnA = lnA₀ - kt, represents the decay of the radioactive isotope lead-214. In this equation, A represents the amount of the substance remaining after time t, A₀ is the original amount of the substance, k is a constant characteristic of the substance, and t is the elapsed time.
To find the amount of lead-214 remaining after 31.6 minutes, we can plug in the given values into the equation. We are given that A₀, the original amount of lead-214 in the sample, is 51.3 mg. The value of k for lead-214 is 2.59×[tex]10^(^-^2^)[/tex][tex]minutes^(^-^1^)[/tex], as mentioned in the question. Finally, t is 31.6 minutes.
Substituting these values into the equation, we have:
lnA = ln(51.3) - (2.59×[tex]10^(^-^2^)[/tex] × 31.6)
Evaluating the right side of the equation, we get:
lnA ≈ 3.937 - (2.59×[tex]10^(^-^2^)[/tex] × 31.6)
≈ 3.937 - 0.8164
≈ 3.1206
To find A, we need to exponentiate both sides of the equation using the natural logarithm base, e:
[tex]e^(^l^n^A^)[/tex] = [tex]e^(^3^.^1^2^0^6^)[/tex]
A ≈ 22.63 mg
Therefore, after 31.6 minutes have passed, approximately 22.63 mg of lead-214 remains in the sample.
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Write "T" for "True" and "F" for "Fulse." Any answer not clearly marked "T or "F" will be counted as incorrect. (1) (5 points) _.Salt water is an example of a mixture becuuse the salt and water can be separated by physical means. (II) (5 points) Coulomb's Law states that the interaction between charges increases as the distance between them increases. (III) (5 points)__The measurement of 101.0 g has three significant figures
The answers are: (I) F, (II) F, (III) T
(I) False. Saltwater is an example of a homogeneous mixture, also known as a solution, where the salt particles are evenly distributed throughout the water. It is not possible to separate salt from water by simple physical means like filtration. To separate the salt from saltwater, a process like evaporation or distillation is required.
(II) False. Coulomb's Law states that the interaction between charges decreases as the distance between them increases. The law describes the electrostatic force between two charged objects, which follows an inverse square relationship. As the distance between charges increases, the force of interaction decreases.
(III) True. The measurement of 101.0 g has three significant figures. In scientific notation, significant figures are the digits that carry meaningful information about the measurement. Non-zero digits and zeros between non-zero digits are considered significant. In this case, all three digits (1, 0, and 1) are non-zero and, therefore, significant.
In summary, saltwater cannot be separated by simple physical means, Coulomb's Law states that the interaction between charges decreases as the distance increases, and the measurement of 101.0 g has three significant figures.
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Calculate the maximum grams of product for the reaction described below by constructing a BCA table and determining the maximum grams of possible product. Complete Parts 1-2 before submitting your answer. Ca3(PO4)2( s)+3H2SO4(aq)→3CaSO4( s)+2H3PO4(aq) A reaction occurs starting with 1.00 kg of Ca3(PO4)2 and 1.00 kg of H2SO4. Based on your knowledge of stoichiometry, set up the table below to determine the amounts of each reactant and product after the reaction goes to completion.. Calculate the maximum grams of product for the reaction described below by constructing a BCA table and determining the maximum grams of possible product. Complete Parts 1−2 before submitting your answer. Ca4(PO4)2( s)+3H2SO4(aq)→3CaSO4( s)+2H3PO4(aq) Based on the table from the previous step, determine the maximum number of grams of CaSOin that can be produced. mass CaSO=
The maximum number of grams of CaSO4 that can be produced is calculated by determining the limiting reactant and using stoichiometry to find the corresponding amount of product.
Which reactant is the limiting reactant in the given reaction?To determine the limiting reactant, we need to compare the moles of each reactant and their stoichiometric ratios in the balanced equation.
1. Calculate the moles of Ca3(PO4)2:
Mass of Ca3(PO4)2 = 1.00 kg = 1000 g Molar mass of Ca3(PO4)2 = (3*40.08 g/mol) + (2*(31.0 g/mol + 4*(16.00 g/mol)))= 310.18 g/mol
Moles of Ca3(PO4)2 = mass/molar mass = 1000 g/310.18 g/mol = 3.22 mol2. Calculate the moles of H2SO4:
Mass of H2SO4 = 1.00 kg = 1000 gMolar mass of H2SO4 = 2*(1.01 g/mol) + 32.07 g/mol + 4*(16.00 g/mol) = 98.09 g/mol Moles of H2SO4 = mass/molar mass = 1000 g/98.09 g/mol = 10.19 mol3. Compare the stoichiometric ratios:
From the balanced equation, the stoichiometric ratio of Ca3(PO4)2 to H2SO4 is 1:3.
The moles ratio of Ca3(PO4)2 to H2SO4 is 3.22 mol : 10.19 mol.
4. Limiting Reactant:
Since the stoichiometric ratio is 1:3, we can see that Ca3(PO4)2 is the limiting reactant because it will be completely consumed before H2SO4.
5. Determine the maximum grams of CaSO4:
The stoichiometric ratio of CaSO4 to Ca3(PO4)2 is 3:1.
The moles of CaSO4 produced will be equal to the moles of Ca3(PO4)2 used.
Moles of CaSO4 = 3.22 mol
Mass of CaSO4 = moles x molar mass = 3.22 mol x (40.08 g/mol) = 129.34 g
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The table shows values for a quadratic function.
x,y
0,0
1,2
2,8
3,18
4,32
5,50
6,72
What is the average rate of change for this function for
the interval from x= 1 to x= 3?
A. 6
B. 4
C. 8
D. 9
The a = 0.Substituting the values of a, b, and c in the general equation, we get:y = 0x² + 2x + 3The quadratic function is:y = 2x + 3Answer: The quadratic function is y = 2x + 3.
The given table illustrates the values of a quadratic function. Here is how you can find the quadratic function:Step 1: Write the general form of a quadratic function y = ax² + bx + c, where y is the dependent variable and x is the independent variable. a, b, and c are constants that affect the shape and position of the parabola.Step 2: Substitute the values from the table for x and y to form a system of equations.Step 3: Solve the system of equations to find the values of a, b, and c. Once you have found these values, substitute them into the quadratic equation to get the quadratic function.
The given table is as follows:x | 0 | 2 | 4 | 6y | 3 | 1 | -1 | -3Step 2:Form a system of equations using the values in the table. Here are the equations:y = a(0)² + b(0) + cy = a(2)² + b(2) + cy = a(4)² + b(4) + cy = a(6)² + b(6) + cStep 3:Solve the system of equations.Using the first equation, y = c. Hence, we have:y = 0²a + 0b + c3 = cThe value of c is 3.Using the second equation, we have:y = 2²a + 2b + 3y = 4a + 2b + 3Subtracting the two equations, we get:- 2a - b = - 2a + b = 2b = 4Therefore, b = 2.Substituting the values of b and c into the first equation, we get:3 = a(0)² + 2(0) + 3
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Use the given conditions to write an equation for the line in
point-slope form and in slope-intercept form.
Slope = - 1/3, passing through (1, - 5)
Given information:Slope = - 1/3Passing through (1, -5)To write the equation of a line in point-slope form, we use the formula as follows:y - y1 = m(x - x1)Where m is the slope and (x1, y1) are the coordinates of the given point.Substituting the given values in the above formula, we have;y - (-5) = -1/3(x - 1)y + 5 = -1/3x + 1/3y = -1/3x + 1/3 - 5y = -1/3x - 14/3Thus, the equation of the line in point-slope form is y - (-5) = -1/3(x - 1) and in slope-intercept form is y = -1/3x - 14/3. Therefore, the equation of the line in point-slope form is y - (-5) = -1/3(x - 1) and in slope-intercept form is y = -1/3x - 14/3.
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A car travel 1/8 mile in 2/13 min what is its speed per min?
The speed of the car is 13/16 mile per minute.
To find the speed of the car per minute, we need to divide the distance traveled by the time taken.
Given that the car traveled 1/8 mile in 2/13 minutes, we can calculate the speed as follows:
Speed = Distance / Time
Speed = (1/8 mile) / (2/13 min)
To divide by a fraction, we can multiply by the reciprocal:
Speed = (1/8 mile) * (13/2 min)
Simplifying the expression:
Speed = (1 * 13) / (8 * 2) mile/min
Speed = 13/16 mile/min
Therefore, the speed of the car is 13/16 mile per minute.
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Find all values of x in the interval [0, 2] that satisfy the equation. (Enter your answers as a comma-separated list.)
18 sin²(x) = 9
Given the equation 18 sin²(x) = 9, the values of x in the interval [0, 2] that satisfy the equation are x = π/4 and x = 3π/4.
To solve the equation, we start by dividing both sides by 18 to isolate the sin²(x):
sin²(x) = 9/18
sin²(x) = 1/2
Next, we use the trigonometric identity sin²θ + cos²θ = 1. By substituting sin²(x) with 1 - cos²(x), we have:
1 - cos²(x) = 1/2
Rearranging the equation, we get:
cos²(x) = 1 - 1/2
cos²(x) = 1/2
Taking the square root of both sides, we have:
cos(x) = ±√(1/2)
cos(x) = ±1/√2
cos(x) = ±√2/2
Since cosine is positive in the first and fourth quadrants, and negative in the second and third quadrants, we have two solutions:
x = π/4 and x = 3π/4
We can confirm that these solutions lie in the interval [0, 2].
Therefore, the values of x in the interval [0, 2] that satisfy the equation are x = π/4 and x = 3π/4.
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Joe rides his bicycle an average of 15 mph. The distance Joe rides d() (in mi) is given by d (1) - 15t, where is the time in hours that he rides. (a) Evaluate d(5) and interpret the meaning. (b) Determine the distance Joe travels in 30 min. Give the exact answer. Do not round. Part: 0/2 Part 1 of 2 (a) d (5)= Thus, Joe travels mi in hours. V
(a) d(5) = 75. Joe travels 75 miles in 5 hours.
(b) Joe travels 7.5 miles in 30 minutes.
(a) Evaluating d(5) means plugging in the value of 5 for t in the equation d(t) = 15t and calculating the resulting distance.
Substituting t = 5 into the equation, we get d(5) = 15 * 5 = 75.
Therefore, d(5) = 75. This means that Joe travels a distance of 75 miles in 5 hours.
(b) We need to determine the distance Joe travels in 30 minutes.
Since the time is given in hours, we convert 30 minutes to hours by dividing by 60 (since there are 60 minutes in an hour): 30 minutes / 60 = 0.5 hours.
Now, we can substitute t = 0.5 into the distance equation: d(0.5) = 15 * 0.5 = 7.5.
Therefore, Joe travels a distance of 7.5 miles in 30 minutes.
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Given a mean and standard deviation of 3,500 and 2,000 cfs, respectively, find the 2-, 10-, and 100-year peak floods for a normal distribution.
The 2-, 10-, and 100-year peak floods for a normal distribution are 5500 cfs.
Given a mean and standard deviation of 3,500 and 2,000 cfs, respectively, the 2-, 10-, and 100-year peak floods for a normal distribution can be found as follows:
Formula used in finding the peak flood is as follows:
Q_T= Q_m + K_T
σ
Where Q_T is the flow for a given period,
Q_m is the mean flow, K_T is the coefficient of skewness, and σ is the standard deviation of the flows.
For a normal distribution, K_T= frac{\text{duration of period in years}-1}{2}\times\frac{\text{duration of period in years}+1}{2}
Substitute the mean and standard deviation to the formula above:
When the period of interest is 2 years, the coefficient of skewness is calculated below:
[{{K}_{T}}=\frac{\text{(2-1)(2+1)}}{2}=1\]
Also, K_{T} is 1 for the 10-year and 100-year flood.
When these values are computed, we get the following values:
Q_{2}=3500+1(2000)=5500 \text{ cfs}
Q_{10}=3500+1(2000)=5500 \text{ cfs}
Q_{100}=3500+1(2000)=5500 \text{ cfs}
Therefore, the 2-, 10-, and 100-year peak floods for a normal distribution are 5500 cfs.
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Evaluate as an integer: 5+3/24-2%1
As an integer the evaluation comes out to be 5 for the expression +3/24-2%1
Expression is: 5 + 3/24 - 2 % 1.
We will solve this expression step by step:
1) we will solve the modulo operation: 2 % 1 = 0.
2) we will solve the division operation: 3/24 = 0.125.
Now, we will substitute the values in the given expression: 5 + 0.125 - 0 = 5.125.
Since we need to evaluate the expression as an integer, we will round it off to the nearest integer.5.125 is closer to 5 than to 6.
Therefore, we will round it down to 5.Hence, the integer value of the given expression is 5.
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Given s(t)=3t
2
+3t, where s(t) is in feet and t is in seconds, find each of the following. a) v(t) b) a(t) c) The velocity and acceleration when t=4sec
The velocity function v(t) is 6t + 3. The acceleration function a(t) is constant and equal to 6. When t = 4 sec, the velocity is 27 ft/s and the acceleration is 6 ft/s^2.
To find the velocity and acceleration, we need to differentiate the position function s(t) with respect to time t.a) Velocity (v(t)): The velocity is the derivative of the position function s(t) with respect to time t.v(t) = d/dt [s(t)]
Given s(t) = 3t^2 + 3t, we can differentiate it to find the velocity:
v(t) = d/dt [3t^2 + 3t]
To differentiate, we apply the power rule of differentiation: v(t) = 6t + 3
Therefore, the velocity function v(t) is 6t + 3.
b) Acceleration (a(t)): The acceleration is the derivative of the velocity function v(t) with respect to time t. a(t) = d/dt [v(t)]
Given v(t) = 6t + 3, we can differentiate it to find the acceleration:
a(t) = d/dt [6t + 3]
The derivative of a constant term is zero, so the derivative of 3 is 0:
a(t) = 6
Therefore, the acceleration function a(t) is constant and equal to 6.
c) Velocity and acceleration when t = 4 sec:
To find the velocity and acceleration at t = 4 seconds, we substitute t = 4 into the respective functions: At t = 4 sec: v(4) = 6(4) + 3
v(4) = 24 + 3
v(4) = 27 ft/s ,a(4) = 6
Therefore, when t = 4 sec, the velocity is 27 ft/s and the acceleration is 6 ft/s^2.
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Evaluate the function \( f(x)=x^{2}+5 x-5 \) at the given values of the independent variable and simplify. a. \( f(-9) \) b. \( f(x+3) \) c. \( f(-x) \) a. \( f(-9)= \) (Simplify your answer.) b. \( f
a. The value of \( f(-9) \) is 63.
b. The simplified form of Function \( f(x+3) \) is \( x²+ 11x + 17 \).
c. The simplified form of \( f(-x) \) is \( x² - 5x - 5 \).
a. To evaluate \( f(-9) \), we substitute -9 for x in the given function:
\( f(-9) = (-9)² + 5(-9) - 5 = 81 - 45 - 5 = 63 \)
b. To evaluate \( f(x+3) \), we substitute \( (x+3) \) for x in the given function:
\( f(x+3) = (x+3)² + 5(x+3) - 5 \)
Expanding and simplifying the expression, we have:
\( f(x+3) = x² + 6x + 9 + 5x + 15 - 5 = x² + 11x + 17 \)
c. To evaluate \( f(-x) \), we substitute -x for x in the given function:
\( f(-x) = (-x)² + 5(-x) - 5 = x² - 5x - 5 \)
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Let sint=a,cost=b, and tant=c. Write the expression in terms of a,b, and c. sin(t+2π)−cos(t+10π)+tan(t+5π)
The given expression in terms of a, b, and c is a - b - c.
Using the trigonometric identities, we can express the given expression in terms of a, b, and c:
sin(t+2π) − cos(t+10π) + tan(t+5π)
Using the periodicity of sine and cosine functions, sin(t+2π) is equal to sin(t) and cos(t+10π) is equal to cos(t). We can substitute these values:
sin(t) − cos(t) + tan(t+5π)
Using the trigonometric identity tan(t+π) = -tan(t), we can rewrite tan(t+5π) as -tan(t):
sin(t) − cos(t) - tan(t)
Now, substituting a for sin(t), b for cos(t), and c for tan(t), we have:
a - b - c
So, A - B - C is the provided phrase in terms of a, b, and c.
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Point P(−8,0) is on the terminal arm of angle θ in standard position. Calculate tanθ. Select one: a. 0 b. −1 c. Undefined. d. 1
Tangent value of zero.
The tangent (tan) of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle containing that angle. In this case, we have a point P(-8, 0) on the terminal arm of angle θ in standard position. Since the x-coordinate is negative and the y-coordinate is zero, we can determine that the point is located on the x-axis, specifically to the left of the origin.
When the y-coordinate is zero, it means that the length of the opposite side of the angle is zero. This implies that there is no vertical displacement from the x-axis. Since the tangent of an angle is defined as the ratio of the opposite side to the adjacent side, and the adjacent side is represented by the x-coordinate,
we have y/x = 0/x = 0.
Therefore, "the tangent of the angle θ at the point P(-8, 0) is zero". This indicates that the angle has no vertical displacement relative to the x-axis. The terminal arm lies entirely on the x-axis, resulting in a tangent value of zero.
In summary, tan θ = 0 because the point P(-8, 0) is situated on the x-axis, to the left of the origin, with no vertical displacement. Division by zero is undefined in mathematics, so the tangent value is zero rather than being undefined.
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If θ=−7π/3, then find exact values for the following. If the
trigonometric function is undefined enter DNE.
Sec
Csc
Tan
Cot
The exact values of the trigonometric functions are:
Sec(θ) = 2
Csc(θ) = (-2√3)/3
Tan(θ) = -√3
Cot(θ) = (-√3)/3
If θ = -7π/3, then the values of the trigonometric functions are as follows:
Sec(θ) = 1/cos(θ) = 1/cos(-7π/3) = 1/(cos(π/3)) = 1/(1/2) = 2
Csc(θ) = 1/sin(θ) = 1/sin(-7π/3) = 1/(sin(-π/3)) = 1/(-√3/2) = -2/√3 = (-2√3)/3
Tan(θ) = sin(θ)/cos(θ) = sin(-7π/3)/cos(-7π/3) = (sin(-π/3))/(cos(-π/3)) = (-√3/2)/(1/2) = -√3
Cot(θ) = 1/tan(θ) = 1/(-√3) = -1/√3 = (-√3)/3
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can you explain how to solve using BAII financial calculator ? 19. Your grandmother has been putting $1000 into a savings account on every birthday since your first (that is,when you turned one.The account pays an interest rate of 3%.How much money will be in the account immediately after your grandmother makes the deposit on your 18th birthday?
The amount of money that will be in the account immediately after your grandmother makes the deposit on your 18th birthday is $29,960.28.
We need to find the amount of money that will be in the savings account after your grandmother makes the deposit on your 18th birthday.
We have the following data:
Present Value (PV) = $1000
Number of payments (N) = 18
Interest rate (I/Y) = 3% per annum.
Using the BAII financial calculator, we can find the future value of this savings account as follows:
1: Clear the calculator. Press the [2nd] [FV] key and then the [C] key.
2: Enter the present value of the savings account. Press the [1000] key and then the [+/-] key.Step 3: Enter the number of payments. Press the [18] key and then the [N] key.
4: Enter the interest rate per period. The interest rate is given in annual terms, so we need to divide it by the number of compounding periods per year. The account pays an interest rate of 3%, so we divide it by 1, since the interest is compounded annually.
Press the [3] key, then the [÷] key, then the [1] key, and finally the [%] key.
5: Calculate the future value. Press the [FV] key. The calculator should display $29,960.28 (rounded to the nearest cent).
Therefore, the amount of money that will be in the account immediately after your grandmother makes the deposit on your 18th birthday is $29,960.28.
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Suppose the function h(x) = sinx is translated StartFraction 3 pi Over 2 EndFraction units left and 11 units down. Which graph represents the result?
Graph D represents the result. The graph D is obtained by shifting the graph of h(x) = sin(x) 3 pi/2 units to the left and 11 units down, which matches the given translation.
The translation of "3 pi/2 units left and 11 units down" implies that each x-coordinate of the original function h(x) = sin(x) is reduced by 3 pi/2, and each y-coordinate is reduced by 11.
Graph D is obtained by shifting the graph of h(x) = sin(x) 3 pi/2 units to the left and 11 units down. This shift is consistent with the given translation. Therefore, Graph D represents the desired result.
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The complete question is:
Suppose the function h(x) = sinx is translated 3pi/2 units left and 11 units down. Which graph represents the result? (Only one graph from below is the correct answer)
Refer to the problem below and do what is required. The iength of a rectangle is 3 times its width. If its length is increased by 2 meters and its width is decreased is 1. meter, the area of the new rectangle is 68 square meter. Find the dimension of the original rectangle.
1. What are the given? 2. What is required? 3. What is the equation that represents the problem?
4. Complete solution:
5. Final answer in complete sentence:
The dimensions of the original rectangle are _______ meters (width) and _______ meters (length). [Please provide the calculated values here.]
1. Given:
- The length of the original rectangle is 3 times its width.
- The area of the new rectangle is 68 square meters after increasing the length by 2 meters and decreasing the width by 1 meter. 2. Required:
- The dimensions of the original rectangle. 3. Equation:
Let's represent the width of the original rectangle as 'w' meters. Then, the length of the original rectangle would be '3w' meters.
The area of a rectangle is calculated by multiplying its length by its width:
Area = Length * Width
4. Solution:
Given that the area of the new rectangle is 68 square meters, we can set up the equation:
(3w + 2) * (w - 1) = 68. Expanding the equation, we get: 3w^2 - w - 70 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. After finding the value of 'w', we can substitute it back into the equation to find the length.
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Solve the following Bernoulli's differential equation: xy - dy/dx = y³ e⁻ˣ^²
The solution to the given Bernoulli's differential equation xy - dy/dx = y³ e⁻ˣ^² is y = -1/(3Ei(-x^2) + C).
To solve the given Bernoulli's differential equation, we can use a substitution. Let's substitute y = u^(1-n) where n is not equal to 0 and 1.
Here's how we solve it step-by-step:
1. Start by differentiating both sides of the equation with respect to x:
d/dx (xy - dy/dx) = d/dx (y^3 e^(-x^2))
2. Simplify the left side using the product rule:
y + x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * d/dx (e^(-x^2))
3. Differentiate the right side using the chain rule:
y + x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * (-2x)
4. Rearrange the equation to isolate dy/dx terms on one side:
x(dy/dx) - dy/dx = 3y^2 e^(-x^2) * (-2x) - y
5. Multiply both sides of the equation by dx:
xdy - ydx = -2x * 3y^2 e^(-x^2) dx - ydx
6. Simplify the equation by canceling out the common terms:
xdy = -6xy^2 e^(-x^2) dx
7. Divide both sides of the equation by x * y^2 to separate variables:
(1/y^2) dy = -6e^(-x^2) dx/x
8. Integrate both sides of the equation:
∫(1/y^2) dy = ∫-6e^(-x^2) dx/x
9. The left side of the equation can be integrated as follows:
∫(1/y^2) dy = -1/y
10. The right side of the equation requires a substitution. Let's substitute u = -x^2, then du/dx = -2x:
∫-6e^(-x^2) dx/x = -6 ∫e^u du/-2u
= 3 ∫e^u du/u
11. The integral on the right side is a special function called the exponential integral, Ei(u). So we have:
-1/y = 3Ei(u) + C
12. Substitute back u = -x^2:
-1/y = 3Ei(-x^2) + C
13. Rearrange the equation to solve for y:
y = -1/(3Ei(-x^2) + C)
That's the solution to the given Bernoulli's differential equation. Remember to consider any initial conditions or constraints to determine the value of the constant C.
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The bearing from City A to City B is N 56° E.The bearing from City B to City C is S 34° E.An automobile driven at 65 mph takes 1.8 hours to drive from City A to City B and takes 1.2hours to drive from City B to City C. Find the distance from City A to City C. (Neglect the curvature of the earth.)
The distance between City A and City C, when an automobile driven at speed of 65 mph takes 1.8 hours to drive from City A to City B and takes 1.2hours to drive from City B to City C, is approximately 287.87 miles.
Given that the bearing from City A to City B is N 56° E and the bearing from City B to City C is S 34° E. Also, the time taken by the automobile to drive from A to B and from B to C is 1.8 hours and 1.2 hours, respectively.
Let's calculate the distance between City A and City B. Let CB = x, then AB = x cosec 56° = x / sin 56°.
Now, let's calculate the distance between City B and City C.BC = x cosec 34° = x / sin 34°
Thus, the distance between City A and City C is AB + BC
Now, AB = x / sin 56° and BC = x / sin 34°. Therefore, the distance between City A and City C = x / sin 56° + x / sin 34° = 110.53 + 177.34 ≈ 287.87 miles (approximately). Therefore, the required distance between City A and City C is approximately 287.87 miles.
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help by solving these questions ... i will mark u as brainliest if u answer correct
The rational numbers in the expressions are: √3 - 2√5, 18/24,19/24,20/24,21/24,and 24/30, 1363636/1000000, 432323/1000000, 59, 60 + 10√2 + 6√3 + √6, = 28 + 12√11
What is surd?Surd is a mathematical concept that refers to a number that cannot be simplified to remove a square root or cube root. For example, 2 is a surd because the square root of 2 cannot be simplified further. Surds are used to simplify calculations and can be used in various fields such as mathematics and physics.
The given parameters are
1) 14/[√108 - √96 +√192 -√54]
14/[6√3 - 4√6 + 8√3 - 3√6]
This implies that
14/ 14√3 -7√6
This is = √3 - 2√5
2) Five rational numbers between 3/4 and 4/5 are
A rational number is a type of real number that can be written as a fraction, where both the numerator and denominator are integers, and the denominator is not equal to zero
These include
Since we want five numbers, we write 3/5 and 4/5 So multiply in numerator and denominator by 5+1 =6 we get
3/4 * 6/6 = 18/24, and 4/5 * 6/6 = 24/30
the rational numbers are
18/24,19/24,20/24,21/24,and 24/30
3_ The value (i) 1.363636 in the form p/q where p and q are integers is
1.363636/1000000
1363636/1000000
(ii) 0.4323232
= 0.432323/1000000
= 432323/1000000
4) (i) (8 + √5)(8+√5)
= 64 - 8√5 + 8√5 -√25
⇒64 -5
= 59
ii) (10 + √3)(6 + √2)
60 + 10√2 + 6√3 + √6
iii) (√3 + √11)² + (√3 + √11)²
Simplifying this to have
(√3 + √11)(√3 + √11) + (√3 + √11)(√3 + √11)
[3 +3√11 + 3√11 + 11] + [3 +3√11 + 3√11 + 11]
= (14 + 6√11) + (14 + 6√11)
= 28 + 12√11
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