The matrix a is diagonalizable if and only if it has n linearly independent eigenvectors, where n is the dimension of the matrix. In this case, a is a 3x3 matrix with diagonal elements 4, 3, and k, and therefore, has three eigenvectors.
To find the eigenvalues, we need to solve the characteristic equation det(a - λI) = 0, where I is the 3x3 identity matrix and λ is the eigenvalue. This yields:
det(a - λI) = (4 - λ)(3 - λ)k - 90 = 0
Expanding and simplifying, we get:
kλ^2 - 7λ^2 + 12λ - 90 = 0
We can factor this quadratic as:
(k - 10)(λ - 6)(λ - 3) = 0
Therefore, the eigenvalues of a are λ = 6, 3, and k - 10. Since a has three linearly independent eigenvectors, it is diagonalizable if and only if all three eigenvalues are distinct. Thus, we need to find the values of k that make the eigenvalues distinct.
If k = 6 or k = 3, then a has repeated eigenvalues and is not diagonalizable. Therefore, the only values of k for which a is diagonalizable are those that make k - 10 ≠ 6 and k - 10 ≠ 3, or k ≠ 16 and k ≠ 13. Thus, the answer is k ≠ 16, 13.
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Solve. Simplify your answer.
log 64
W =
W =
Submit
1
6
The simplified value of log 64 (with base 10) is approximately 2.5.
To solve the logarithm equation log 64, we need to determine the base of the logarithm. Assuming the base is 10 (common logarithm), we can rewrite the equation as: log₁₀ 64
The logarithm function asks the question: "To what power must we raise the base (10) to obtain the given number (64)?" In this case, we need to find the exponent that produces 64 when the base 10 is raised to that power.
To simplify, we recall that 10 to the power of 2 is equal to 100:
10² = 100
Similarly, 10 to the power of 3 is equal to 1000:
10³ = 1000
Since 64 is between 10² and 10³, we can conclude that the exponent will be between 2 and 3. We can estimate that the exponent is closer to 2.5.
Thus, the simplified value of log 64 (with base 10) is approximately 2.5.
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3. A random sample of students were surveyed as to how much non-school screen time they had each week
and if their grade average was above or below 80.
What PERCENT of students who spend 4-8 hrs
average above 80. Round your answer to the nearest
The number of students who for between 4-8 hours and obtained an average above 80 expressed as a percentage is 11.7%.
Calculating PercentagesRather than expressing values in fractions. A certain portion of a whole lot or item can be multiplied by 100 to get its equivalent value expressed as a percentage .
From the table , the number of students who studied for 4-8 hours and also had a grade above 80 is 11.
Total number of students in the sample = 94
Expressing as a percentage;
(11/94) × 100%
= 0.117 × 100%
= 11.7%
Hence, the percentage value is 11.7%
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find the ordered pair that corresponds to the given pair of parametric equations and value of t. x=second power 5, y=4 -third power; t=3
Thus, the ordered pair corresponding to x= (t^2) + 5, y= 4 - (t^3), and t=3 is (14,-23).
To find the ordered pair corresponding to the given pair of parametric equations and the value of t, we need to substitute t=3 into the equations for x and y and simplify.
x= (t^2) + 5
x= (3^2) + 5 = 14
y= 4 - (t^3)
y= 4 - (3^3) = -23
Therefore, the ordered pair that corresponds to the given pair of parametric equations and the value of t=3 is (14,-23).
Parametric equations are equations that express a set of variables as functions of one or more independent variables, called parameters. In this case, x and y are expressed as functions of the parameter t. Parametric equations are often used in physics, engineering, and other fields where there are variables that depend on time or other independent variables.
In summary, to find the ordered pair corresponding to a given pair of parametric equations and a specific value of t, we substitute t into the equations for x and y and simplify to obtain the values of x and y at that point. In this example, the ordered pair corresponding to x= (t^2) + 5, y= 4 - (t^3), and t=3 is (14,-23).
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The radius of a circle is 5 kilometers. What is the angle measure of an arc 3 kilometers long?
Find d/dx integral from 2^(3 x+1) ln (t+1) dt. show work please
The derivative of the given function can be found using the Fundamental Theorem of Calculus and the chain rule as follows:
d/dx integral from 2^(3 x+1) ln (t+1) dt
= d/dx [integral from a(x) to b(x) f(t) dt] (where a(x) = 1 and b(x) = 2^(3x+1) and f(t) = ln(t+1))
= f(b(x)) * b'(x) - f(a(x)) * a'(x)
= ln(2^(3x+1) + 1) * (2^(3x+1) * ln(2)) - ln(2) * 1
= ln(2) * 2^(3x+1) * (3x + 1) * ln(2^(3x+1) + 1)
Therefore, the derivative of the given function is ln(2) * 2^(3x+1) * (3x + 1) * ln(2^(3x+1) + 1).
The above solution was obtained by applying the Fundamental Theorem of Calculus, which states that the derivative of the definite integral of a function f(t) with respect to x is given by f(b(x)) * b'(x) - f(a(x)) * a'(x), where a(x) and b(x) are functions of x that define the limits of integration. The chain rule was used to compute b'(x), which is the derivative of 2^(3x+1) with respect to x. Finally, the derivative of the integrand ln(t+1) with respect to x was computed using the chain rule.
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LIn makes a line plot to show the data in the table. He places one dot above the 2 on the scale. How many dots should he place above the 3?
The number of dots that should be place above the 3 in the dot plot is: 4 dots
How to Interpret Dot Plots?A dot plot is one that is used to represent any data in the form of dots or small circles. It is similar to a simplified histogram or a bar graph as the height of the bar formed with dots represents the numerical value of each variable. Dot plots are thus used to represent small amounts of data. For example, a dot plot can be used to collect the vaccination report of newborns in an area, which is represented in the following table.
Now, from the given table, we see the pea pods and the number of peas they have.
Now, from the table, only one Pea pod has 2 peas and that's why we have one dot above 2.
However, we can see that 4 pea pods have 4 number of peas and as such we will have 4 dots above 3.
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Find the equation of the tangent line at the indicated point. (Use symbolic notation and fractions where needed.)R(z)=log7(2z2−151),z=10
The equation of the tangent line to R(z) at z = 10 is:
y = (40/(ln 7 * 49))x - (390/(ln 7 * 49))
To find the equation of the tangent line to the function R(z) = log7(2z^2 - 151) at z = 10, we first need to find the derivative of R(z) with respect to z:
R'(z) = d/dz [log7(2z^2 - 151)]
= 1/(ln 7) * 1/(2z^2 - 151) * d/dz[2z^2 - 151] (by the chain rule)
= 1/(ln 7) * 1/(2z^2 - 151) * 4z
= 4z/(ln 7 * (2z^2 - 151))
Now we can evaluate R'(10) to find the slope of the tangent line at z = 10:
R'(10) = 4(10)/(ln 7 * (2(10)^2 - 151))
= 40/(ln 7 * 49)
So the slope of the tangent line at z = 10 is 40/(ln 7 * 49).
Next, we need to find the y-coordinate of the point on the graph of R(z) that corresponds to z = 10. We can do this by evaluating R(10):
R(10) = log7(2(10)^2 - 151)
= log7(249)
Therefore, the point on the graph of R(z) that corresponds to z = 10 is (10, log7(249)).
Finally, we can use the point-slope form of the equation of a line to write the equation of the tangent line:
y - log7(249) = (40/(ln 7 * 49))(x - 10)
Simplifying this equation gives:
y = (40/(ln 7 * 49))x + log7(249) - (40/(ln 7 * 49)) * 10
So the equation of the tangent line to R(z) at z = 10 is:
y = (40/(ln 7 * 49))x - (390/(ln 7 * 49))
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What are the coordinates of the midpoint of the segment whose endpoints are A(-1,-2) and B(6,12)?
o (-3, 18)
o (5, 10)
o (7, 14)
o (2.5, 5)
The coordinates of the midpoint of the line segment AB are (2.5, 5).
The correct answer is: o (2.5, 5)
To find the midpoint of the line segment with endpoints A(-1, -2) and B(6, 12), we can use the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Here, x1 and y1 are the coordinates of point A, and x2 and y2 are the coordinates of point B.
Plugging in the values, we get:
Midpoint = ((-1 + 6) / 2, (-2 + 12) / 2)
= (5 / 2, 10 / 2)
= (2.5, 5)
Therefore, the coordinates of the midpoint of the line segment AB are (2.5, 5).
The correct answer is:
o (2.5, 5)
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The coordinates of the midpoint are (2.5, 5). So, the correct answer is (2.5, 5).
To find the coordinates of the midpoint of the segment with endpoints A(-1, -2) and B (6,12), we can use the midpoint formula. The midpoint formula states that the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the y-coordinate of the midpoint is the average of the y-coordinates of the endpoints.
Let's apply the midpoint formula:
x-coordinate of the midpoint = (x-coordinate of A + x-coordinate of B) / 2
= (-1 + 6) / 2
= 5 / 2
= 2.5
y-coordinate of the midpoint = (y-coordinate of A + y-coordinate of B) / 2
= (-2 + 12) / 2
= 10 / 2
= 5
Therefore, this means that the midpoint of the segment with endpoints A(-1,-2) and B(6,12) is located at the coordinates (2.5, 5). The x-coordinate represents the average of the x-values of the endpoints, and the y-coordinate represents the average of the y-values of the endpoints.
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find a value for h so that the equation ax = 0 has a nonzero solution x, where a = [1 -1 2 1 0 h 2 -1 2]
To find a value for h so that the equation ax = 0 has a nonzero solution x, we need to determine the null space of matrix a. The null space is the set of all solutions x that satisfy the equation ax = 0. If the null space contains a nonzero vector, then we have found a value for h that satisfies the condition.
To find the null space, we row reduce the augmented matrix [a|0]. After performing row operations, we obtain:
[1 -1 0 3 0 h-1 0 1 0|0]
From this, we can see that the third and sixth variables are free, and we can express the other variables in terms of these. Setting h = -2, we can find a nonzero solution for x. For example, letting the third and sixth variables be 1 and 0 respectively, we get:
x = [1, -1, 2, -1, 0, 1, 0, 1, 2]
Therefore, a value of h = -2 will give a nonzero solution to the equation ax = 0.
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find ℒ{f(t)} by first using a trigonometric identity. (write your answer as a function of s.) f(t) = 12 cos t − 6
The Laplace transform of f(t) = 12 cos t - 6 is ℒ{f(t)} = 12s / (s^2 + 1) - 6/s.
To find the Laplace transform ℒ{f(t)} of the function f(t) = 12 cos t - 6, we can apply the Laplace transform property involving the cosine function and a trigonometric identity. The property states:
ℒ{cos(at)} = s / (s^2 + a^2)
Using this property, we can split the Laplace transform into two parts:
ℒ{f(t)} = ℒ{12 cos t} - ℒ{6}
Applying the Laplace transform property to each term:
ℒ{12 cos t} = 12 * ℒ{cos t} = 12 * (s / (s^2 + 1^2)) = 12s / (s^2 + 1)
ℒ{6} = 6 * ℒ{1} = 6 * (1 / s) = 6/s
Combining the two terms, we have:
ℒ{f(t)} = 12s / (s^2 + 1) - 6/s
Therefore, the Laplace transform of f(t) = 12 cos t - 6 is ℒ{f(t)} = 12s / (s^2 + 1) - 6/s.
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the demonstration of a statistical relationship between scores on a predictor and scores on a criterion measure is called:
The demonstration of a statistical relationship between scores on a predictor and scores on a criterion measure is called criterion-related validity.
This type of validity refers to the extent to which a test, assessment, or measurement tool can accurately predict or correlate with an established outcome, such as performance in a particular job or success in a specific academic setting.
There are two types of criterion-related validity: predictive validity and concurrent validity. Predictive validity evaluates how well the predictor scores forecast future criterion performance, while concurrent validity assesses the relationship between predictor and criterion scores at the same time.
Establishing criterion-related validity involves correlating the scores on a predictor measure, such as a standardized test, with the scores on a criterion measure, like job performance ratings or academic achievement. A significant correlation indicates that the predictor measure has the ability to predict or estimate the criterion measure, thus demonstrating criterion-related validity.
In summary, criterion-related validity is crucial in determining the effectiveness and relevance of a test, assessment, or measurement tool by evaluating the statistical relationship between predictor and criterion scores. This helps ensure that the predictor measure serves its intended purpose and accurately reflects the desired outcomes.
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Determine if the following statement is true or false. To perform a one-way ANOVA, the populations do not need to be normally distributed. This statement is false or true?
The statement "To perform a one-way ANOVA, the populations do not need to be normally distributed" is true. One-way ANOVA (analysis of variance) is a statistical test used to determine whether there are significant differences between the means of three or more groups.
It is based on the assumption that the populations being compared have equal variances and that the observations are independent and identically distributed.
However, ANOVA does not require the populations to be normally distributed, but rather the residuals (i.e., the differences between the observed values and the predicted values) should be normally distributed. This means that the sample sizes for each group should be large enough to satisfy the central limit theorem, which states that the means of samples taken from any population will be approximately normally distributed if the sample size is large enough.
Therefore, while it is ideal for the populations to be normally distributed, it is not a requirement for performing a one-way ANOVA. Other assumptions, such as homogeneity of variances, independence of observations, and equal sample sizes, should also be met to ensure the validity of the results.
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The given figure is a right triangular prism. The volume is 210in. In the prism, JL=7 inches and KM is equal to 6 inches. What is the length of JN?
Answer:
JN = 10 in
Step-by-step explanation:
the volume (V) of a triangular prism is calculated as
V = Ah ( A is the area of the triangular base and h the height )
A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the perpendicular height )
here b = JL = 7 , h = KM = 6 , then
A = [tex]\frac{1}{2}[/tex] × 7 × 6 = [tex]\frac{1}{2}[/tex] × 42 = 21 in²
given V = 210 with h = JN , then
21 JN = 210 ( divide both sides by 21 )
JN = 10 in
help please, which answer is it ?
Answer:
<1 and <4
Step-by-step explanation:
Adjacent means "next to". Only 1 and 4 are next to each other.
Find the value of x.
The value of x is given by the following option:
E. 64º.
How to obtain the value of x?To obtain the value of x, we must consider that the sum of the internal angle measures of a triangle is of 180º.
The exterior angle theorem states that an exterior angle is supplementary with it's respective interior angle, hence the second interior angle of the triangle has the measure given as follows:
<A + 96º = 180º
<A = 84º.
Hence the value of x is obtained as follows:
x + 32 + 84 = 180
x + 116 = 180
x = 64º.
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The value of x in the attached image is 66°
How to solve for the value of xThere are many approach to solving for the value of x in the diagram:
One of them is using the exterior angle property to find the interior angleUsing the sum of the angles in triangleTo use the first approach, we apply the Exterior Angle Property. Exterior Angle Property states that an exterior angle of a triangle is equal to the sum of its two opposite non-adjacent interior angles.
we are given:
∠ABC = 32°
∠DAB = 98° (exterior angle of a triangle)
∠ACB = x°
By applying the exterior angle property, then we can have the equation:
∠DAB = ∠ABC + ∠ACB
Substitute the values into the above equation:
98 = 32 + x
make x the subject of the formula
x = 98 - 32
x = 66°
Therefore the value of x is 66°
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suppose x, the years of learning a second language of a student, is a normal distribution random variable with mean of 7 years and standard deviation of 2.5 years. what is the probability that a student learns more than 11 years?
The probability that a student learns more than 11 years is approximately 0.0548 or 5.48%.
What is probability?Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence.
To find the probability that a student learns more than 11 years, we need to standardize the variable using the standard normal distribution. We can do this by calculating the z-score for 11 years as follows:
z = (x - μ) / σ
z = (11 - 7) / 2.5
z = 1.6
Here, μ is the mean of the distribution (7 years) and σ is the standard deviation (2.5 years). We have calculated the z-score as 1.6.
We can now use a standard normal distribution table or a calculator to find the probability that a z-score is greater than 1.6. The probability of a z-score being greater than 1.6 is approximately 0.0548.
Therefore, the probability that a student learns more than 11 years is approximately 0.0548 or 5.48%.
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Find x and y for the problem.
By factorials, the values of variables x and y are 2 and 1560, respectively.
How to find the result of a product of fractions
In this problem we need to find the values of the variables x and y derived from the multiplication of 38 fractions, whose definition is done by the following expression involving factorials:
n! / [(n + 2)! / 2!] = x / y
2 · n! / (n + 2)! = x / y
2 / [(n + 1) · (n + 2)] = x / y
If we know that n = 38, then the values of x and y are, respectively:
x = 2
y = 39 · 40
y = 1560
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John drove from station A to station B a distance of 224 miles. On his way back he increased his speed by 10 mph. If the journey back took him 24 min less, what was his original speed?
Answer:
70 mph------------------
Find the time in travel in both directions, show their difference, considering the distance is 224 and the original speed is s:
224/s - 224/(s + 10) = 24 (time in minutes)224/s - 224/(s + 10) = 24/60 (time in hours)224/s - 224/(s + 10) = 2/5Multiply both sides by 5s(s + 10)/2 and simplify to get quadratic equation:
s² + 10s - 5600 = 0s² + 80s - 70s - 5600 = 0s(s + 80) - 70(s + 80) = 0(s - 70)(s + 80) = 0s = 70 and s = - 80The second root is negative and hence is discarded, hence the answer is 70 mph.
if t is a linear transformation, then t(0) enter your response here and for all vectors u, v in the domain of t and all scalars c, d.
If we let u=0 and v= 0, we have t(0+0)=t(0)+t(0), which implies t(0)=0.
If t is a linear transformation, then t(0)=0 for any vector space.
This is because the zero vector is the additive identity of any vector space and by the definition of a linear transformation, t(u+v) = t(u) + t(v) for any vectors u and v in the domain of t.
Thus, if we let u=0 and v= 0, we have t(0+0)=t(0)+t(0), which implies t(0)=0.
Intuitively, this means that a linear transformation does not change the location of the origin. Geometrically, it implies that the image of the origin under a linear transformation is also the origin.
This property is important in many areas of mathematics, such as linear algebra and differential equations, where linear transformations are used to study the behavior of functions and systems.
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What is the circumference of a circle with a radius of 94. 2 using 3. 14 for pi
Answer:
591.88
Step-by-step explanation:
C = 2πr
2π(94.2)
591.8760559
round to the nearest ones, tenths, or hundredths (depends on your question)
i did hundredth↓
591.88
Taylor made a pattern of perfect squares. She had 16, 25, 36, ____, 64, 81 in her pattern. What number needs to be squared to find the missing number? what is the answer please
Answer:
he answer is 49.
Step-by-step explanation:
To find the missing number in the pattern of perfect squares, we can observe that the given numbers are arranged in increasing order. The missing number should fit the pattern of perfect squares.
The given numbers are: 16, 25, 36, ____, 64, 81.
The pattern suggests that each number is the square of a certain integer. Let's find the missing number by looking at the square root of each given number:
√16 = 4,
√25 = 5,
√36 = 6,
_____,
√64 = 8,
√81 = 9.
From the above calculations, we can see that the missing number is the square of 7, since √49 = 7. Therefore, the missing number in the pattern is 49.
So, the answer is 49.
Answer choices
It is positive and increasing
It is positive and decreasing
It is negative and increasing
It is negative and decreasing
Answer:
A
Step-by-step explanation:
the graph goes up and the value of y increases
what is the answer to the math problem
-15.4+25.2+(-10.4)=
find the volume generated by rotating the region bounded by y = ln ( x ) , the x-axis and the vertical line x = e 2 about the x-axis. express your answer in exact form.
The volume generated by rotating the region bounded by y = ln ( x ) is V = π(xₐ³ln(xₐ) - (xₐ²/2)) - 4πe⁶ln(e²) + e⁴/2
To find the volume generated by rotating the region about the x-axis, we'll divide the region into infinitely thin vertical strips, and then rotate each strip around the x-axis to form a cylindrical shell.
Using the formula for the volume of a cylindrical shell, we have:
V = ∫(2πrh)dx
To express r and h in terms of x, we observe that r is simply x (the distance from the x-axis to the strip), and h is ln(x) (the height of the strip). Substituting these values, we have:
V = ∫(2πx * ln(x))dx
To evaluate this integral, we can use integration by parts. Let's assign u = ln(x) and dv = 2πx dx. Then, du = (1/x) dx and v = πx². Applying integration by parts, we get:
V = [u * v] - ∫(v * du)
= [ln(x) * πx²] - ∫(πx² * (1/x) dx)
= πx³ln(x) - π∫(x dx)
= πx³ln(x) - π(x²/2) + C
where C is the constant of integration.
Now, we need to evaluate this expression at the upper and lower limits of x. Recall that the lower limit is e², and the upper limit is xₐ (which is a variable). So, the volume V becomes:
V = π(xₐ³ln(xₐ) - (xₐ²/2)) - π(e²)³ln(e²) + (e²)²/2
Since we want to express the answer in exact form, we'll leave it in terms of xₐ. Hence, the volume generated by rotating the given region about the x-axis is:
V = π(xₐ³ln(xₐ) - (xₐ²/2)) - 4πe⁶ln(e²) + e⁴/2
This expression represents the volume in exact form.
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find the volume of the solid that lies inside both the cylinder x 2 y 2 = 1 x2 y2=1 and the sphere x 2 y 2 z 2 = 4 x2 y2 z2=4
The volume of the solid that lies inside both the cylinder x^2 + y^2 = 1 and the sphere x^2 + y^2 + z^2 = 4 can be found using triple integrals.
The cylinder x^2 + y^2 = 1 is centered at the origin and has a radius of 1, while the sphere x^2 + y^2 + z^2 = 4 is also centered at the origin and has a radius of 2. To find the volume of the solid inside both surfaces, we need to integrate over the region of overlap between the two shapes.
The region of overlap is defined by the cylinder along the z-axis, which intersects the sphere at z = ±√3. Therefore, we can integrate over the region where -√(4 - x^2 - y^2) ≤ z ≤ √(4 - x^2 - y^2), x^2 + y^2 ≤ 1, and obtain the volume of the solid:
∫∫∫ dv = ∫∫∫ dz dA
Using cylindrical coordinates, we have:
∫∫∫ dz dA = ∫0^2π ∫0^1 ∫-√(4-r^2)^(√r^2) r dz dr dθ
Evaluating this triple integral yields:
V = 8π/3 - 4√3π/3
Therefore, the volume of the solid that lies inside both the cylinder x^2 + y^2 = 1 and the sphere x^2 + y^2 + z^2 = 4 is 8π/3 - 4√3π/3.
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For a moving object, the force acting on the object varies directly with the object's acceleration. When a force of 30 N acts on a certain object, the acceleration
of the object is 3 m/s. If the forde is changed to 50 N, what will be the acceleration of the object?
Answer:
the acceleration of the object will be 5 m/s^2 when the force is 50 N.
Step-by-step explanation:
The force acting on the object varies directly with the object's acceleration, so we can use the formula:
force = constant x acceleration
where the constant is the same for both situations.
We can solve for the constant by plugging in the given values:
30 = constant x 3
constant = 10
Now we can use the constant to find the acceleration when the force is 50 N:
50 = 10 x acceleration
acceleration = 5 m/s^2
Therefore, the acceleration of the object will be 5 m/s^2 when the force is 50 N.
Factor the polynomial, if possible. Drag the expressions into the box if they are part of the factored form of the polynomial. If the polynomial cannot be factored, drag prime. 6x^2+4x−16
The expressions to drag into the box are 2, 3x-4, and x+2.
To factor the polynomial [tex]6x^2+4x-16[/tex], we can first factor out the greatest common factor, which is 2:
[tex]2(3x^2 + 2x - 8)[/tex]
Then we can factor the quadratic expression inside the parentheses:
2(3x-4)(x+2)
So the factored form of the polynomial is:
2(3x-4)(x+2)
Therefore, the expressions to drag into the box are:
2, 3x-4, and x+2.
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Tank A holds 300 gallons of water and it has been
filled with water at a rate of 5 gallons per hour.
Tank B holds 348 gallons of water and it is leaking
3 gallons per hour. In how many hour both tanks
will hold the same amiunt of water?
a) 4 hours
b) 3 hours
c) 6 hours
d) 7 hours
Answer:
Step-by-step explanation:
After calculating the rate at which water is being filled in Tank A and the rate at which water is being leaked from Tank B, it can be determined that both tanks will hold the same amount of water after 4 hours. Therefore, the correct answer is option a) 4 hours.
normalize the following vectors. (a) u = 13i − 6j 8k, v = i 7j − k
The normalized vector v is:
v = (1/√51)i + (7/√51)j + (-1/√51)k
To normalize a vector, we need to divide it by its magnitude. The magnitude of a vector v = (v₁, v₂, v₃) is given by:
|v| = √(v₁² + v₂² + v₃²)
(a) To normalize u = 13i − 6j + 8k, we first need to calculate its magnitude:
|u| = √(13² + (-6)² + 8²) = √(169 + 36 + 64) = √269
Then, we can normalize u by dividing it by its magnitude:
u = u / |u| = (13/√269)i + (-6/√269)j + (8/√269)k
Therefore, the normalized vector u is:
u = (13/√269)i + (-6/√269)j + (8/√269)k
(b) To normalize v = i + 7j − k, we first need to calculate its magnitude:
|v| = √(1² + 7² + (-1)²) = √51
Then, we can normalize v by dividing it by its magnitude:
v = v / |v| = (1/√51)i + (7/√51)j + (-1/√51)k
Therefore, the normalized vector v is:
v = (1/√51)i + (7/√51)j + (-1/√51)k
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The vertices of a triangle are A(6,2) , B(-2,0) , and C(-4,2) . Draw the image after a dilation with a scale factor of 1/2
After a dilation with a scale factor of 1/2, the new vertices of the triangle are A'(3,1), B'(-1,0), and C'(-2,1). The dilated triangle is a smaller version of the original triangle, maintaining the same shape and proportions.
To draw the image of the triangle after a dilation with a scale factor of 1/2, we need to calculate the new coordinates of each vertex. Here's a detailed human-generated answer without plagiarism:
Given vertices:
A(6, 2)
B(-2, 0)
C(-4, 2)
Dilation with scale factor of 1/2:
To dilate the triangle, we will multiply the x and y coordinates of each vertex by the scale factor (1/2).
New coordinates calculation:
A'(x, y) = (1/2 × 6, 1/2 × 2) = (3, 1)
B'(x, y) = (1/2 × -2, 1/2 × 0) = (-1, 0)
C'(x, y) = (1/2 × -4, 1/2 × 2) = (-2, 1)
New coordinates:
A'(3, 1)
B'(-1, 0)
C'(-2, 1)
Now, let's plot the original triangle and the image after dilation:
Original Triangle:
A(6, 2)
B(-2, 0)
C(-4, 2)
Dilated Triangle:
A'(3, 1)
B'(-1, 0)
C'(-2, 1)
Here's the graphical representation of the original triangle (solid lines) and the dilated triangle (dashed lines):
In the diagram, the original triangle is represented by solid lines (connecting vertices A, B, and C), and the dilated triangle is represented by dashed lines (connecting vertices A', B', and C'). The dilation with a scale factor of 1/2 has resulted in the triangle being reduced in size by half.
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