Answer:
56.25 feet.
Step-by-step explanation:
h(t) = 60t - 16t^2
Differentiating to find the velocity:
v(t) = 60 -32t
This equals zero when the ball reaches its maximum height, so
60-32t = 0
t = 60/32 = 1.875 seconds
So the maximum height is h(1.875)
= 60* 1.875 - 16(1.875)^2
= 56.25 feet.
Answer: 56.25 feet.
Step-by-step explanation:
For a Quadratic function in the form [tex]f(x)=ax^2+bx+c[/tex], if [tex]a<0[/tex] then the parabola opens downward.
Rewriting the given function as:
[tex]h(t) = - 16t^2+60t[/tex]
You can identify that [tex]a=-16[/tex]
Since [tex]a<0[/tex] then the parabola opens downward.
Therefore, we need to find the vertex.
Find the x-coordinate of the vertex with this formula:
[tex]x=\frac{-b}{2a}[/tex]
Substitute values:
[tex]x=\frac{-60}{2(-16)}=1.875[/tex]
Substitute the value of "t" into the function to find the height in feet that the ball will reach. Then:
[tex]h(1.875)=- 16(1.875)^2+60(1.875)=56.25ft[/tex]
All tangents to the circle are congruent and form a square. The perimeter of square ACEG is 24 cm. What is the length of line segment BC?
Answer:
3 cm
Step-by-step explanation:
The perimeter of the square is 24 cm, so the side length is 6 cm.
I assume B is the point between A and C where the tangent line intersects the circle. If so, B is the midpoint of AC, so it is half the length. Therefore, BC = 3 cm.
The length of line segment BC is 3 cm.
What is a Square ?A square is a polygon with four sides , all the sides of the square are equal.
It is given that the tangents of the circle are forming a square ,
The perimeter of the square is 24 cm.
the length of line segment BC = ?
The perimeter of the square is 4a
where a is the side of the square.
Substituting the values
24 = 4 * a
a = 6 cm
B is the mid point of the tangent length and therefore
BC = 6 /2 = 3 cm
Therefore , the length of line segment BC is 3 cm.
The missing image is attached with the answer.
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Find the inverse of the matrices if they Exist. Use the algorithm introduced in the section. [1 0 -2 -3 1 4 2 -3 4]
Answer:
\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right]
Step-by-step explanation:
Given is a matrix 3x3 as
[tex]\left[\begin{array}{ccc}1&0&2\\-3&1&4\\2&-3&4\end{array}\right][/tex]
|A| =2 hence inverse exists.
Cofactors are 16 20 7
6 8 3
2 2 1
Hence inverse =
[tex]\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right][/tex]
It is given that y is directly proportional to x.
What is the constant of variation?
Answer:
The constant of variation is -4
Step-by-step explanation:
If y is directly proprtional to x, then we write:
[tex]y \propto \: x[/tex]
If we introduce the constant of proportionality or variation k, then we obtain:
[tex]y = kx[/tex]
When x=1, y=-4 (We can choose any ordered pair from the table)
[tex] - 4 = k(1)[/tex]
[tex]k = - 4[/tex]
Therefore the constant of variation is -4.
Answer:
-4
Step-by-step explanation:
Find the following: F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k Exercise Find the curl and the divergence of the vector field.
[tex]\vec F(x,y,z)=e^{xy}\sin z\,\vec\jmath+y\tan^{-1}\dfrac xz\,\vec k[/tex]
Divergence is easier to compute:
[tex]\mathrm{div}\vec F=\dfrac{\partial(e^{xy}\sin z)}{\partial y}+\dfrac{\partial\left(y\tan^{-1}\frac xz\right)}{\partial z}[/tex]
[tex]\mathrm{div}\vec F=xe^{xy}\sin z-\dfrac{xy}{x^2+z^2}[/tex]
Curl is a bit more tedious. Denote by [tex]D_t[/tex] the differential operator, namely the derivative with respect to the variable [tex]t[/tex]. Then
[tex]\mathrm{curl}\vec F=\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\D_x&D_y&D_z\\0&e^{xy}\sin z&y\tan^{-1}\frac xz\end{vmatrix}[/tex]
[tex]\mathrm{curl}\vec F=\left(D_y\left[y\tan^{-1}\dfrac xz\right]-D_z\left[e^{xy}\sin z\right]\right)\,\vec\imath-D_x\left[y\tan^{-1}\dfrac xz\right]\,\vec\jmath+D_x\left[e^{xy}\sin z}\right]\,\vec k[/tex]
[tex]\mathrm{curl}\vec F=\left(\tan^{-1}\dfrac xz-e^{xy}\cos z\right)\,\vec\imath-\dfrac{yz}{x^2+z^2}\,\vec\jmath+ye^{xy}\sin z\,\vec k[/tex]
To find the curl and divergence of a given vector field, you first identify the vector's components. The curl is calculated using a determinant and the divergence is obtained by computing a dot product of the gradient operator with the vector field.
Explanation:In order to find the curl and the divergence of the vector field F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k, we first need to identify its components. The components are as follows: e^(xy) sin z, and y tan^−1(x/z).
The Curl of a vector field F in three dimensions is typically denoted as ∇ × F or curl F, where '∇' is the del operator. In Cartesian coordinates, this can be calculated using a determinant that involves the unit vectors î, ĵ, and k, the gradient operator, and the components of F.
The Divergence of a vector field F in three dimensions, typically denoted as ∇ . F or div F, is obtained by computing a dot product of the gradient operator with the vector field. This can also be calculated using Cartesian coordinates.
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You deposit $400 each month into an account earning 5% interest compounded monthly. a. How much will you have in the account in 30 years? b. How much total money will you put into the account? c. How much total interest will you earn?
a) After 30 years, the amount in the account will be approximately $499,355.18.
b) The total money deposited over 30 years will be $144,000.
c) The total interest earned over 30 years will be approximately $355355.18.
We have,
Given:
Monthly deposit: $400
Interest rate: 5% (expressed as a decimal, 0.05)
Time: 30 years (in months, 30 * 12 = 360 months)
a.
To calculate the amount in the account after 30 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Amount in the account
P = Initial deposit (first deposit)
r = Annual interest rate
n = Number of times interest is compounded per year
t = Time in years
In this case:
P = $400 x 360 = $144000
r = 0.05
n = 12 (compounded monthly)
t = 30
Substituting the values into the formula:
A = 144000(1 + 0.05/12)^(12 * 30)
A ≈ $499,355.18
b.
The total money deposited can be calculated by multiplying the monthly deposit by the number of months:
Total Money Deposited = Monthly deposit * Number of months
Total Money Deposited = $400 * 360
Total Money Deposited = $144,000
c.
The total interest earned can be calculated by subtracting the total money deposited from the amount in the account:
Total Interest Earned = Amount in the account - Total Money Deposited
Total Interest Earned = $499,355.18 - $144,000
Total Interest Earned ≈ $355355.18
Therefore,
After 30 years, the amount in the account will be approximately $499,355.18.
The total money deposited over 30 years will be $144,000.
The total interest earned over 30 years will be approximately $355355.18.
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To find the future value of the account after 30 years, use the compound interest formula. Multiply the monthly deposit by the number of months to find the total money put into the account. The total interest earned is found by subtracting the total money put into the account from the future value.
Explanation:To calculate the future value of the account, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the future value, P is the principal (initial deposit), r is the interest rate in decimal form, n is the number of times interest is compounded per year, and t is the number of years.
a. Plugging in the values, we have [tex]A = 400(1 + 0.05/12)^(12*30)[/tex]. Using a calculator, the future value after 30 years will be approximately $1000.40.
b. To find the total money put into the account, we multiply the monthly deposit by the number of months. In this case, it will be $[tex]400 * 12 * 30 = $144,000.[/tex]
c. The total interest earned can be found by subtracting the total money put into the account from the future value. In this case, it will be $[tex]1000.40 - $144,000 = -$143,999.60.[/tex]
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Identify the radius and center.
x^2 + y^2 - 8x - 6y + 16 = 0
The answer is:
Center: (4,3)
Radius: 3 units.
Why?To solve the problem, using the given formula of a circle, we need to find its standard equation form which is equal to:
[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]
Where,
"h" and "k"are the coordinates of the center of the circle and "r" is its radius.
So, we need to complete the square for both variable "x" and "y".
The given equation is:
[tex]x^2+y^2-8x-6y+16=0[/tex]
So, solving we have:
[tex]x^2-8x+y^2-6y=-16[/tex]
[tex](x^2-8x+(\frac{8}{2})^{2} )+(y^2-6y+(\frac{6}{2})^{2})=-16+(\frac{8}{2})^{2}+(\frac{6}{2})^{2}\\\\(x^2-8x+16)+(y^2-6y+9)=-16+16+9\\\\(x^2-4)+(y^2-3)=9[/tex]
Now, we have that:
[tex]h=4\\k=3\\r=\sqrt{9}=3[/tex]
So,
Center: (4,3)
Radius: 3 units.
Have a nice day!
Note: I have attached a picture for better understanding.
By completing the square on the given equation, the center of the circle is found to be (4, 3) and the radius is 3.
This process will transform the equation into a standard form of a circle equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius of the circle.
To start, we can rewrite the original equation by adding and subtracting the necessary constants inside the square terms:
x^2 - 8x can become (x - 4)^2 by adding and then subtracting 16 inside the equation.y^2 - 6y can become (y - 3)^2 by adding and then subtracting 9 inside the equation.Therefore, after completing the square, we get (x - 4)^2 + (y - 3)^2 = 1. Hence, the center of the circle is (4, 3) and the radius is 3.
1) Draw the arrow diagram and the matrix representation for the relation: R={(1, 2), (3, 4), (2, 3), (3, 2), (2, 1), (3, 1), (4, 3)} with domain {1, 2, 3, 4)
Answer:
The arrow diagram and the matrix representation for the relation is shown below.
Step-by-step explanation:
The given relation is
R={(1, 2), (3, 4), (2, 3), (3, 2), (2, 1), (3, 1), (4, 3)}
If a relation is defined as
[tex]R=\{(x,y)|x\in R,y\in R\}[/tex]
Then the set of x values is domain and set of y values is range.
The domain of the function is
Domain={1, 2, 3, 4)
The range of the function is
Range={1, 2, 3, 4)
In arrow diagram, we have two sets first set represents the domain and second set represents the range. The arrow connecting the element represent the relation.
In matrix representation,
[tex]M_{ij}=\begin{cases}1 & \text{ if } (x_i,y_j)\in R \\ 0 & \text{ if } (x_i,y_j)\notin R\end{cases}[/tex]
The arrow diagram and the matrix representation for the relation is shown below.
The arrow diagram and matrix representation offer effective visualizations to understand and communicate the relationships within the given relation R={(1, 2), (3, 4), (2, 3), (3, 2), (2, 1), (3, 1), (4, 3)}.
The given relation R={(1, 2), (3, 4), (2, 3), (3, 2), (2, 1), (3, 1), (4, 3)} can be analyzed step by step. The domain, representing the set of x values, is Domain={1, 2, 3, 4}, and the range, representing the set of y values, is Range={1, 2, 3, 4}.
The arrow diagram visually represents this relation, with the first set depicting the domain and the second set depicting the range. Arrows connect elements to illustrate the relationships within the given set.
The matrix representation further encapsulates this relation, with rows corresponding to the elements of the domain and columns to the elements of the range. The presence of an entry in the matrix indicates a relation between the respective elements.
This method provides a concise and organized representation of the given relation. Overall, the arrow diagram and matrix representation serve as effective tools to comprehend and communicate the relationships within the specified set.
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Find an equation of the line that passes through the given points. (Let x be the independent variable and y be the dependent variable.) (3, 9) and (4, 14)
Answer:
y=5x-6
Step-by-step explanation:
Hello
let´s see , we have P1(3,9) and P2(4.14)
first let's find its slope
[tex]m=\frac{y2-y1}{x2-x1} = \frac{14-9}{4-3}[/tex]
[tex]m=\frac{5}{1} \\m=5[/tex]
the slope is 5.
[tex]y-y_{0} = m(x-x_{0} )\\\\using P1\\\\ y-9 = 5(x-3)\\\\y=5x-15+9\\\\y=5x-6[/tex]
where x is the independent variable and y is the dependent variable
Have a great day.
To find the equation of the line that passes through (3, 9) and (4, 14), calculate the slope and use the point-slope form, resulting in the equation y = 5x - 6 in slope-intercept form.
To find an equation of the line that passes through the points (3, 9) and (4, 14), we will first calculate the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1). Substituting the given points into the formula, we have m = (14 - 9) / (4 - 3) = 5. With the slope known, we can use one of the points and the slope to write the equation in point-slope form, which is y - y1 = m(x - x1). Using the point (3, 9), the equation becomes y - 9 = 5(x - 3). To put this into slope-intercept form, we simplify to get y = 5x - 6.
(From Textbook) A company makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. If chicken costs 10 cents per ounce and grain costs 1 cent per ounce, how many ounces of each should the company use in each bag of dog food in order to minimize cost?
Answer:
The company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.
Step-by-step explanation:
Let x be the number of ounces of chicken
let y be the number of ounces of grains
We are given that Chicken has 10 grams of protein and 5 grams of fat per ounce.
Chicken has protein in 1 ounce = 10
Chicken has protein in x ounces = 10x
Chicken has fat in 1 ounce = 5
Chicken has fat in x ounces = 5x
We are also given that grain has 2 grams of protein and 2 grams of fat per ounce
Grain has protein in 1 ounce = 2
Grain has protein in y ounces = 2y
Grain has fat in 1 ounce = 2
Grain has fat in y ounces =2y
Now we are given that . A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat.
So, equation becomes:
[tex]10x+2y\geq 200[/tex] ---A
[tex]5x+2y\geq 150[/tex] ---B
Since x and y represent the ounces .
So, [tex]x\geq 0[/tex] ---C and [tex]y\geq 0[/tex] ----D
Plot A ,B ,C and D on the graph
Refer the attached graph so , the corner points are (0,100),(10,50) and (30,0)
We are also given that chicken costs 10 cents per ounce and grain costs 1 cent per ounce
So, Cost function becomes : [tex]c=10x+y[/tex]
At(0,100)
[tex]c=10x+y[/tex]
[tex]c=10(0)+100[/tex]
[tex]c=100[/tex]
At(10,50)
[tex]c=10x+y[/tex]
[tex]c=10(10)+50[/tex]
[tex]c=150[/tex]
At(30,0)
[tex]c=10x+y[/tex]
[tex]c=10(30)+0[/tex]
[tex]c=300[/tex]
Minimum cost is 100 cents at (0,100)
So, the company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.
Find the equation of the line passing through (4, 3) and (-2, -7): A. 3x + 5y = 27 B. 3x - 5y = -3 C. 5x - 3y = 11 D. 2x - y = 5
Answer:
(c) 5x - 3y = 11
Step-by-step explanation:
using slope intercept form
we know equation of line is
y= mx+c........(1)
where m is slope which can be written as [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{-7-3}{-2-4} =\frac{-10}{-6} =\frac{5}{3}[/tex]
substituting value in equation (1)
[tex]y=\frac{5}{3} x+c[/tex]
now inserting value of point (4,3)
[tex]3=\frac{5}{3}\times 4+c[/tex]
[tex]c=\frac{-11}{3}[/tex]
[tex]y=\frac{5}{3} \times x-\frac{11}{3}\\[/tex]
on solving we get
[tex]5x - 3y = 11[/tex]
use logarithmic differentiation to find dy/dx
y=(lnx)^x
Try this suggested solution.
4. Without performing the divisions, determine whether the integers 176521221 and 149235678 are divisible by 9 or 11. - Obtain the following aralization . The int e nted
Final answer:
By using the divisibility rules for 9 and 11, we can conclude that the integer 176521221 is divisible by 9 but not by 11, and the integer 149235678 is divisible by 9 but not by 11.
Explanation:
To determine whether the integers 176521221 and 149235678 are divisible by 9 or 11 without performing the division, we can use divisibility rules:
Divisibility by 9: Add up all the digits in the number. If the sum is divisible by 9, then the original number is also divisible by 9.
Divisibility by 11: Take the alternating sum of the digits. If the result is divisible by 11 (including 0), the original number is divisible by 11.
Divisibility of 176521221 by 9
1+7+6+5+2+1+2+2+1 = 27; Since 27 is divisible by 9, 176521221 is also divisible by 9.
Divisibility of 176521221 by 11
(1-7+6-5+2-1+2-2+1) = -3; Since -3 is not divisible by 11, 176521221 is not divisible by 11.
Divisibility of 149235678 by 9
1+4+9+2+3+5+6+7+8 = 45; Since 45 is divisible by 9, 149235678 is also divisible by 9.
Divisibility of 149235678 by 11
(1-4+9-2+3-5+6-7+8) = 9; Since 9 is not divisible by 11, 149235678 is not divisible by 11.
A decade-old study found that the proportion of high school seniors who felt that "getting rich" was an important personal goal was 72% . Suppose that we have reason to believe that this proportion has changed, and we wish to carry out a hypothesis test to see if our belief can be supported. State the null hypothesis and the alternative hypothesis that we would use for this test.
Answer: [tex]H_0:p\leq0.72[/tex]
[tex]H_a:p\neq0.72[/tex]
Step-by-step explanation:
Given : A decade-old study found that the proportion of high school seniors who felt that "getting rich" was an important personal goal was 72% .
Let 'p' be the new proportion of high school seniors who felt that "getting rich" was an important personal goal .
Claim : [tex]p\neq0.72[/tex]
We know that the null hypothesis has equal sign.
Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-
[tex]H_0:p=0.72[/tex]
And the alternative hypothesis must be :-
[tex]H_a:p\neq0.72[/tex]
Hence, the null hypothesis and the alternative hypothesis that we would use for this test :
[tex]H_0:p\leq0.72[/tex]
[tex]H_a:p\neq0.72[/tex]
Final answer:
To test the belief that the proportion of high school seniors valuing wealth has changed, we would use H0: p = 0.72 as the null hypothesis and Ha: p
Explanation:
To conduct a hypothesis test regarding the proportion of high school seniors who believe that "getting rich" is an important goal, we state the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis is the statement that the proportion remains the same, while the alternative hypothesis states that the proportion has changed.
For a proportion of high school seniors who believe that getting rich is an important goal, if the decade-old study showed a proportion of 72%, we would have:
H0: p = 0.72
Ha: p ≠ 0.72
In the hypothetical scenario with a disease prevalence of 9.5% in the general population and finding 7% in a local survey, we would determine the null and alternative hypotheses regarding whether the local proportion is less than the national average as follows:
H0: p ≥ 0.095
Ha: p < 0.095
When conducting a hypothesis test, the null hypothesis typically represents no change or no effect, while the alternative hypothesis represents a change, difference, or effect that we are trying to detect.
Solve the initial value problem (explicit solution). y y' - cot t = 0 y(pi/2) = -1
This ODE is separable:
[tex]yy'-\cot t=0\implies y\,\mathrm dy=\cot t\,\mathrm dt[/tex]
Integrate both sides to get
[tex]\dfrac12y^2=\ln|\sin t|+C[/tex]
Given that [tex]y\left(\frac\pi2\right)=-1[/tex], we get
[tex]\dfrac12(-1)^2=\ln|\sin\dfrac\pi2\right|+C\implies C=\dfrac12[/tex]
Then
[tex]\dfrac12y^2=\ln|\sin t|+\dfrac12[/tex]
[tex]y^2=2\ln|\sin t|+1[/tex]
[tex]y^2=\ln\sin^2t+1[/tex]
[tex]\implies\boxed{y(t)=\pm\sqrt{\ln\sin^2t+1}[/tex]
Find and identify the traces of the quadric surface x2 + y2 − z2 = 36 given the plane. x = k Find the trace. Incorrect: Your answer is incorrect. Identify the trace. circle ellipse hyperbola parabola y = k
Answer:
Hyperbola.
Step-by-step explanation:
We start from the function:
[tex]x^{2} +y^{2} -z^{2} =36[/tex]
We may get the traces if we cut the surface in planes x=k. This is equivalent to replace x for k:
[tex]k^{2}+y^{2} -z^{2}=36\\y^{2} -z^{2}=36-k^{2}[/tex]
The last equation is the equation of an hyperbola, which varies its characteristics depending on K (the plane cut).
When x=k or y=k is substituted into the equation of the quadric surface x² + y² - z² = 36, the resulting traces are hyperbolas. These traces represent intersections of planes parallel to coordinate planes with the quadric surface.
Explanation:The trace of a quadric surface can be found by substituting a constant into the equation for one of the variables. The given quadric surface equation is x² + y² - z² = 36. If we substitute x = k into the equation, we get k² + y² - z² = 36. This is an equation of a hyperbola.
Similarly, for the case when y = k, we substitute this into our original equation to get x² + k² - z² = 36. This also represents a hyperbola.
The traces given by these equations represent the intersection of planes parallel to the coordinate planes and the quadric surface. In these particular cases, the traces are hyperbolas.
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Use a truth table to determine which of the following is true for this proposition:
(p & (q v p)) → ~p
Question 1 options:
The proposition is never true.
The proposition is false when p is false.
The proposition is false if p is true.
The proposition is always true.
Answer:
The proposition is false if p is true.
Step-by-step explanation:
I made the truth table in the picture, to fill it you need to know:
v : only is false when both propositions are false.
^ : only is true when both propositions are true.
⇒: only is false when the left proposition is true and the right proposition is false.
So, the answer is the third option.
The truth table analysis shows that the proposition (p & (q \/ p)) → ~p is a tautology, meaning it is always true regardless of the truth values of p and q.
To determine the truth value of the proposition (p & (q \/ p)) \/ ~p, we need to construct a truth table and evaluate the proposition for all possible truth values of p and q.
Truth Table Construction
Let's go step by step:
Construct two initial columns for the variables p and q with their possible truth values.
Create a column for the sub-proposition (q \/ p), which is true if either q or p (or both) is true.
Construct the column for the main proposition (p & (q \/ p)) which is true if both p is true and the previous sub-proposition is true.
Determine the negation of p (~p), which is true when p is false.
Finally, combine the results to evaluate the column for the entire proposition (p & (q \/ p)) \/ ~p, using the implication operator (→), whose result is only false when the antecedent is true and the consequent is false.
After evaluating, we find that the proposition is always true, making it a tautology. Therefore, the original proposition (p & (q \/ p)) → ~p is always true irrespective of the truth values of p and q.
What is the equation of the line that is tangent to the circle of radius 8 at (0,-8) and whose center is at the origin? Click to select your answer 2 3 4
Answer:
y=-8
Step-by-step explanation:
So if you draw a circle with center at the origin (0,0) with radius 8.
So we have:
That means the radius stretches to (0,-8); down 8 units from (0,0).
The radius stretches to (0,8); up 8 units from (0,0).
The radius stretches to (8,0); right 8 units from (0,0).
The radius stretches to (-8,0); left 8 units from (0,0).
We are looking for a line tangent to our circle at (0,-8). Since this was down 8 units. Then our equation is horizontal and y=a number. They-coordinate in (0,-8) is -8, so the we have y=-8.
If someone said tangent at (0,8), we would have said y=8 .
If someone said tangent at (8,0), we would have said x=8.
If someone said tangent at (-8,0), we have have said x=-8.
The equation of the line that is tangent to the circle with radius 8 at (0,-8) and center at the origin is y=-8. This is because the tangent line is horizontal at that point as it is perpendicular to the radius line lying along the y-axis.
Explanation:In the context of Mathematics, particularly geometry, the line that is tangent to a circle at a given point is perpendicular to the radius drawn to that point. Since the circle's center is at the origin (0,0) and the radius is extended to the point (0,-8), this radius lies along the y-axis. Therefore, the equation of the tangent line which is perpendicular to this radius would be a horizontal line through the point (0,-8), given by the equation y=-8.
Remember that tangent line by definition touches the circle at only one point without intersecting it and the line is perpendicular to the radius at that point of tangency. In this particular scenario, the radius and tangent are perpendicular lines in the coordinate system: the radius aligns with the y-axis, hence the tangent aligns with the x-axis.
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Please help me with this
Answer:
Yes, the triangles are congruent by Hypotenuse angle congruence
Step-by-step explanation:
we know that
The Hypotenuse-Angle Congruence states that If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the triangles are congruent.
In this problem
The hypotenuse and an acute angle of the right triangle in the left are congruent to the hypotenuse and corresponding acute angle of the right triangle in the right
therefore
The triangles are congruent by Hypotenuse angle congruence
An exam worth 269 points contains 36 questions. Some questions are worth 7 points, and the others are worth 8 points. How many 7 point and 8 point questions are on the test?
Answer:
There are 19 questions worth 7 points.
There are 17 questions worth 8 points.
Step-by-step explanation:
Assign variables:
Let x = number of questions worth 7 points.
Let y = number of questions worth 8 points.
First we deal with the number of questions.
There are 36 questions, so our first equation is:
x + y = 36
Now we deal with the points.
x questions worth 7 points each are worth 7x points.
y questions worth 8 points each are worth 8y points.
The total worth of all the questions is 7x + 8y.
The total worth of the exam is 269, so our second equation is
7x + 8y = 269
We have a system of 2 equations in 2 variables.
x + y = 36
7x + 8y = 269
Let's use the substitution method to solve the system of equations.
Solve the first equation for x.
x + y = 36
x = 36 - y
Substitute 36 - y in for x in the second equation.
7x + 8y = 269
7(36 - y) + 8y = 269
Distribute the 7.
252 - 7y + 8y = 269
Combine like terms.
252 + y = 269
Subtract 252 from both sides.
y = 17
There are 17 questions worth 8 points.
x + y = 36
x + 17 = 36
x = 19
There are 19 questions worth 7 points.
Check:
What does 19 questions at 7 points each and 17 questions at 8 points each add to?
19 * 7 + 17 * 8 = 133 + 136 = 269
The points add to 269, so our answer is correct.
determine if it is possible to form a triangle using the segments with the given measurements
42) 8ft, 9ft, 11ft
43) 7.4cm, 8.1cm, 9.8cm
EXPLAIN YOUR REASONING!
Answer:
Part 42) Yes, It is possible o form a triangle using the segments with the given measurements
Part 43) Yes, It is possible o form a triangle using the segments with the given measurements
Step-by-step explanation:
we know that
The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side
Part 42) we have
8ft, 9ft, 11ft
Applying the triangle inequality theorem
a) 8+9 > 11
17 ft > 11 ft ----> is true
b) 9+11 > 8
20 ft > 8 ft ----> is true
c) 8+11 > 9
19 ft > 9 ft ----> is true
therefore
Yes, It is possible o form a triangle using the segments with the given measurements
Part 43) we have
7.4cm, 8.1cm, 9.8cm
Applying the triangle inequality theorem
a) 7.4+8.1 > 9.8
15.5 cm > 9.8 cm ft ----> is true
b) 8.1+9.8 > 7.4
17.9 cm > 7.4 cm ----> is true
c) 7.4+9.8 > 8.1
17.2 cm > 8.1 cm ----> is true
therefore
Yes, It is possible o form a triangle using the segments with the given measurements
According to the label, a can of soup holds an average of 307 grams, with a standard deviation of 4.1 grams. Assuming a normal distribution, what is the probability that a can will be sold that holds more than 308 grams?
Answer: 0.4052
Step-by-step explanation:
Given : Mean : [tex]\mu=\text{307 grams}[/tex]
Standard deviation : [tex]\sigma = \text{4.1 grams}[/tex]
The formula for z -score :
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x=308 ,
[tex]z=\dfrac{308-307}{4.1}=0.24390\approx0.24[/tex]
The p-value = [tex]P(z>0.24)=1-P(z<0.24)[/tex]
[tex]=1-0.5948348= 0.4051652\approx0.4052[/tex]
Hence, the probability that a can will be sold that holds more than 308 grams =0.4052.
PLEASE HELP
Write equations for the vertical and horizontal lines passing through the point (-6, -5)
Answer:
see below
Step-by-step explanation:
A horizontal line has the same y value and is a constant y value
y = -5
A vertical line has the same x value and is a constant x value
x = -6
A vertical line is a line where all of the [tex]x[/tex] values are the same. In this case, [tex]\boxed{x=-6}[/tex], so that is the equation of the line.
A horizontal line is a line where all of the [tex]y[/tex] values are the same. Here, [tex]\boxed{y=-5}[/tex], so that is the second line.
A study to determine the sensitivity and specificity of a new test for macular degeneration is conducted on 2430 people. Macular degeneration occurs at a rate of 16.72%. Your sample has the same prevalence of macular degeneration. You find that 377 people with macular degeneration tested positive with the new test. You also have a total of 561 positive test results in your study. CALCULATE THE SPECIFICITY of this test.
Question 2 options:
A)
83.29%
B)
98.45%
C)
92.86%
D)
67.20%
E)
90.91%
A rectangle has a length 10 m less than twice its width. When 2 m are added to the width, the resulting figure is a square with an area of 196 m squared. Find the dimensions of the original rectangle.
Answer:
The length of the original rectangle is 14 meters and the width of the original rectangle is 12 meters
Step-by-step explanation:
Let
l ----> the length of the original rectangle
w ----> the width of the original rectangle
we know that
[tex]l=2w-10[/tex] ----> equation A
[tex]196=(w+2)^{2}[/tex] ----> area of a square
Solve for w
square root both sides
[tex](w+2)=(+/-)14[/tex]
[tex]w=14=(+/-)14-2[/tex]
[tex]w=14=14-2=12\ m[/tex]
[tex]w=14=-14-2=16\ m[/tex] -----> this solution not make sense
so
[tex]w=12\ m[/tex]
Find the value of L
[tex]l=2(12)-10=14\ m[/tex]
therefore
The length of the original rectangle is 14 meters and the width is 12 meters
Final answer:
The dimensions of the original rectangle are 12 meters (width) and 14 meters (length).
Explanation:
To find the dimensions of the original rectangle, let's first set up an equation.
Let the width of the rectangle be x meters.
The length of the rectangle is given as 10 m less than twice its width, so the length is (2x - 10) meters.
When 2 m are added to the width, the resulting figure is a square with an area of 196 m2.
The side length of a square with area A is √A, so the side length of the square is √196 = 14 meters.
Since adding 2 m to the width creates a square with side length 14 meters, we have:
x + 2 = 14
Subtracting 2 from both sides gives:
x = 12
Therefore, the dimensions of the original rectangle are 12 meters (width) and
(2*12 - 10)
= 14 meters (length).
Michelle wants to order cookies. It is $6 for 4 pcs, how much would it cost for 20 pcs?
6 divided by 4 is 1.5.
1.5 times 20 is 30
it would cost michelle $30 to buy 20 pieces of cookies
Answer:
$30
Step-by-step explanation:
4pcs = $6
20pcs = x
Cross multiplying, we have
x = (20pcs * $6)/4pcs
= $120/4 = $30
Jerry King is a server in a restaurant that pays a salary of $43 per day. He also averages tips of 18% of his total gross food orders. Last week he worked 4 days and had total food orders of $2,312.5. What was his total gross pay for the week? Round intermediate calculations and final answer to the nearest cent 416.25
Answer:
His total gross pay for the week is $588.25.
Step-by-step explanation:
Consider the provided information.
Jarry has a salary of $43 per day and he worked for 4 days.
Thus, the salary of 4 days is:
4 × $43= $172
The average tips is 18% of his total gross food orders.
Last week he had total food orders of $2,312.5. Therefore, the total tips he received is:
[tex]2312.5 \times(\frac{18}{100})[/tex]
[tex]2312.5 \times(0.18)[/tex]
[tex]416.25[/tex]
Total tips he received is $416.25.
His total gross pay for the week = The salary of 4 days + Total tips he received.
His total gross pay for the week = $172 + $416.25
His total gross pay for the week = $588.25
Hence, his total gross pay for the week is $588.25.
Final answer:
Jerry King's total gross pay for the week, combining his salary and the tips he made on his food orders, is $588.25, after calculating an 18% tip on his total orders and adding his daily salary for the 4 days he worked.
Explanation:
The question asks us to calculate the total gross pay for Jerry King, a server who earns both a salary and tips based on his food orders. First, we need to calculate Jerry's earnings from tips. He averages an 18% tip on his total gross food orders. Since his total food orders for the week were $2,312.5, we calculate the tips as follows:
Tips = (Total Food Orders) × (Tip Percentage)
= $2,312.5 × 18%
= $2,312.5 × 0.18
= $416.25 (rounded to the nearest cent)
Next, we calculate Jerry's salary for the week. Jerry works 4 days per week at a salary of $43 per day:
Salary = (Daily Salary) × (Number of Days Worked)
= $43 × 4
= $172
Finally, to find Jerry's total gross pay for the week, we add the salary and the tips:
Total Gross Pay = Salary + Tips
= $172 + $416.25
= $588.25 (rounded to the nearest cent)
Therefore, Jerry King's total gross pay for the week is $588.25.
Write an equation of the circle with center (9, -3) and radius 8.
Answer:
(x-9)^2 + (y+3)^2 = 8^2
or
(x-9)^2 + (y+3)^2 = 64
Step-by-step explanation:
An equation for a circle can be written as
(x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r is the radius
(x-9)^2 + (y- -3)^2 = 8^2
(x-9)^2 + (y+3)^2 = 8^2
or
(x-9)^2 + (y+3)^2 = 64
Determine whether S is a basis for P3. S = {4t - 12,5 +t3,5 +3t,-3t2 +2/3
Answer:
Yes , S is a basis for [tex]P_3[/tex].
Step-by-step explanation:
Given
S=[tex]\left\{4t-12,5+t^3,5+3t,-3t^2+\frac{2}{3}\righ\}[/tex].
We can make a matrix
Let A=[tex]\begin{bmatrix}-12&4&0&0\\5&0&0&1\\5&3&0&0\\\frac{2}{3}&0&-3&0\end{bmatrix}[/tex]
All rows and columns are linearly indepedent and S span [tex]P_3[/tex].Hence, S is a basis of [tex]P_3[/tex]
Linearly independent means any row or any column should not combination of any rows or columns.
Because a subset of V with n elements is a basis if and only if it is linearly independent.
Basis:- If B is a subset of a vector space V over a field F .B is basis of V if satisfied the following conditions:
1.The elements of B are linearly independent.
2.Every element of vector V spanned by the elements of B.
A car's velocity changes at a rate of 36 miles per hour in 0.5 hour.
What is its acceleration, in miles/hour2?
(Note: Do not specify units in your answer. Your answer must be only a number.)
Answer:
72
Step-by-step explanation:
In order to find the acceleration we need first to defined it, so:
acceleration=((final velocity)-(initial velocity))/time interval
[tex]a=(vf-vi)/dt[/tex]
But (vf-vi) actually represents the velocity change, so (vf-vi)/dt represents the velocity change rate. This means that in our case:
[tex]a=(36miles/hour)/(0.5hours)[/tex]
[tex]a=72miles/hour^2[/tex]
In conclusion the acceleration is [tex]a=72miles/hour^2[/tex], without units just 72.
I need help with the following question.
It is about Math of Finance.
Questions:
A $3500 loan was settled ten years later with a payment of $8100. Determine the annual rate of return.
Answer:
The annual rate of return is 13.14%.
Step-by-step explanation:
As it is not mentioned whether the amount was compounded, so we will assume this to be simple interest.
Given is - A $3500 loan was settled ten years later with a payment of $8100.
Means total amount paid back was = $8100
And original principle was = $3500
So, interest paid = [tex]8100-3500=4600[/tex] dollars
Now simple interest formula is :
[tex]I=p\times r\times t[/tex]
Where p = 3500
I = 4600
r = ?
t = 10
Now putting these values in formula we get;
[tex]4600=3500\times r\times10[/tex]
[tex]r=4600/35000[/tex]
r= 0.1314
So, rate of interest = 13.14%