Answer:
○ Length = 20 centimeters and Width = 5 centimeters
Step-by-step explanation:
Let the length = L
Then the width is L - 15
A = lw
100 = L(L - 15)
L² - 15L - 100 = 0
(L - 20)(L + 5) = 0
L - 20 = 0 or L + 5 = 0
L = 20 or L = -5
Since a length of a rectangle cannot be negative, we discard the solution L = -5 and keep the solution L = 20.
L = 20
width = L - 15 = 20 - 15 = 5
Answer: ○ Length = 20 centimeters and Width = 5 centimeters
7. What is the ratio of the interior angles of a pentagon and a decagon?
Answer:
A pentagon has 5 sides.
Sum of the interior angles = ( n - 2) x 180
Sum of the interior angles = ( 5 - 2) x 180
Sum of the interior angles = 540°
STEP 2: Find one interior angle:
one interior angle = 540 ÷ 5 = 108°
STEP 3: Find the sum of interior angles of a decagon
A decagon has 10 sides
Sum of the interior angles = ( n - 2) x 180
Sum of the interior angles = ( 10 - 2) x 180
Sum of the interior angles = 1440°
STEP 4: Find one interior angle:
one interior angle = 1440 ÷ 10 = 144°
STEP 5: Find the ratio:
Pentagon : Decagon = 108 : 144
Simplify:
Pentagon : Decagon = 3 : 4
Answer: The ratio is 3 : 4
Please help me solve this
A function assigns the values. The value of (p·p)(x) is x⁴ + 10x³ + 30x² + 25x.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
Given p(x)= x²+5x, therefore, the value of (p·p)(x) can be written as,
(p·p)(x) = p(p(x))
= (x²+5x)²+5(x²+5x)
= x⁴ + 25x² + 10x³ + 5x² + 25x
= x⁴ + 10x³ + 30x² + 25x
Hence, the value of (p·p)(x) is x⁴ + 10x³ + 30x² + 25x.
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Write an expression that can be used to produce vector b from vector a, and explain how you determined that expression.
Answer:
Expression that can be used to produce vector b from vector a
= T× magnitude of vector a × vector 'a'
Step-by-step explanation:
Given question is based on the concept of vectors
Vectors:- Vectors is a geometric entity that has both magnitude and direction. Vectors have an initial point at the point where they start and a terminal point that tells the final position of the point. Various operations can be applied to vectors such as addition, subtraction, and multiplication
To produce an another vector form a given vector we increase its magnitude only not its direction as we have to produce another vector 'b' with increased magnitude from a given vector 'a'.
its clear from the question that only the magnitude of vector a is increased to produce another vector b
hence we multiply any constant ('T') with the magnitude of vector a to increase its magnitude.
and finally multiply the increased magnitude with the vector 'a' itself to get the answer.
so, the final expression that can be used to produce vector b from vector a is = T× magnitude of vector a × vector 'a'.
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Use Cramer Rule to solve the following system: 8x−5y=70 and 9x+7y=3
Answer:
[tex](x,y) = (5,-6)[/tex]
Step-by-step explanation:
[tex]\underline{\textbf{Determinant of a matrix.}}\\\\\text{For a}~ 2 \times 2 ~ \text{matrix,}\\\\\begin{vmatrix} a_1&a_2\\b_1&b_2 \end{vmatrix} = a_1b_2 - a_2b_1\\\\\\\text{For a}~ 3 \times 3 ~ \text{matrix,}\\\\\begin{vmatrix} a_1&a_2&a_3\\ b_1&b_2&b_3\\ c_1&c_2&c_3 \end{vmatrix} = a_1\begin{vmatrix} b_2&b_3\\c_2&c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1&b_3\\c_1&c_3 \end{vmatrix}+ a_3 \begin{vmatrix} b_1&b_2\\c_1&c_2 \end{vmatrix}\\\\\\[/tex]
[tex]~~~~~~~~~~~~~~~~~~=a_1(b_2c_3-b_3c_2) -a_2(b_1c_3-b_3c_1) +a_3(b_1c_2-b_2c_1)[/tex]
[tex]\underline{\textbf{Cramer's Rule to solve a system of two equations.}}\\\\\text{Consider the system of two equations:}\\\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a_1x + b_1 y= c_1\\\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a_2x +b_2 y = c_2\\\\\text{Here,}\\\\x = \dfrac{D_x}{D}= \dfrac{\begin{vmatrix} c_1&b_1\\c_2&b_2 \end{vmatrix}}{\begin{vmatrix} a_1&b_1\\a_2&b_2 \end{vmatrix}}\\\\\\ y= \dfrac{D_y}{D}= \dfrac{\begin{vmatrix} a_1&c_1\\a_2&c_2 \end{vmatrix}}{\begin{vmatrix} a_1&b_1\\a_2&b_2 \end{vmatrix}}\\\\[/tex]
[tex]\underline{\textbf{Solution:}}\\\\~~~~~~~~~~~~~~~~~~~~~~~8x-5y = 70~~~~~~...(i)\\\\~~~~~~~~~~~~~~~~~~~~~~~9x +7y = 3~~~~~~~...(ii)\\\\\text{Applying Cramer's rule:}\\\\x = \dfrac{D_x}{D}\\\\\\~~=\dfrac{\begin{vmatrix} 70& -5 \\3&7 \end{vmatrix}}{\begin{vmatrix} 8& -5\\ 9& 7\end{vmatrix}}\\\\\\~~=\dfrac{70(7) -(-5)(3)}{(8)(7)-(-5)(9)}\\\\\\~~=\dfrac{490+15}{56+45}\\\\\\~~=\dfrac{505}{101}\\\\\\~~=5[/tex]
[tex]y = \dfrac{D_y}{D}\\\\\\~~=\dfrac{\begin{vmatrix} 8& 70 \\9&3 \end{vmatrix}}{\begin{vmatrix} 8& -5\\ 9& 7\end{vmatrix}}\\\\\\~~=\dfrac{(8)(3) -(70)(9)}{(8)(7)-(-5)(9)}\\\\\\~~=\dfrac{24-630}{56+45}\\\\\\~~=-\dfrac{606}{101}\\\\\\~~=-6[/tex]
[tex]\textbf{Hence, the solution to the system of equation is}~ (x,y) = (5,-6)[/tex]
Pls ,see the attachment
the polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at x=2 and x=0, and a root of multiplicity at x=-2.
find a possible formula for P(x)
P(x)=
Polynomial is an expression that consists of indeterminates(variable) and coefficients. The possible formula for P(x) is x⁵ - 2x⁴ - 4x³ + 8x².
What are polynomials?Polynomial is an expression that consists of indeterminates(variable) and coefficient, it involves mathematical operations such as addition, subtraction, multiplication, etc, and non-negative integer exponentials.
Given the roots of the polynomial, therefore, the following details can be written as,
The polynomial has a root of multiplicity 2 at x=2, (x-2)²The polynomial has a root of multiplicity 2 at x=0, (x-0)²The polynomial has a root of multiplicity 1 at x=-2, (x+2)¹Therefore, the polynomial will become,
P(x) = (x-2)²(x-0)²(x+2)¹
= (x²+4-4x) (x)² (x+2)
= (x²+4-4x) (x³+2x²)
= x⁵ + 2x⁴ + 4x³ + 8x² - 4x⁴ -8x³
= x⁵ - 2x⁴ - 4x³ + 8x²
Hence, the possible formula for P(x) is x⁵ - 2x⁴ - 4x³ + 8x².
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a) Students returning to a secondary school have to pass through a check point at the rate of 50 per though. The average time for checking is 35 seconds. The arrival rate and service rate follow a Poisson distribution. There is a delay and the principal is willing to include another check point to reduce the average time of checking to 28 seconds if the idle time is less than 10% and the average queue length is more than 10 students, check whether the introduction of an extra check point is justifiable
Answer:
175
Step-by-step explanation:
Average queue length = (arrival rate * service time) / (1 - arrival rate * service time)
For the given problem, the average queue length is:
Average queue length = (50 * 35) / (1 - 50 * 35)
= 175
Which degenerate conic is formed when a double cone is sliced through the a pex by a plane parallel to the slant edge of the cone?
Circle
Parabola
One line
Two lines
One line will be formed which will be parallel to the slant edge option third is correct.
What is a conic section?It is defined as the curve which is the intersection of cone and plane. There are three major conic sections; parabola, hyperbola, and ellipse (circle is a special type of ellipse).
We have a statement:
Which degenerate conic is formed when a double cone is cut through the top by a plane parallel to the slant edge of the cone?
As we can see in the attached picture if the double cone is cut through the top by a plane parallel to the slant edge of the cone one line will be formed which will be parallel to the slant edge.
Thus, one line will be formed which will be parallel to the slant edge option third is correct.
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f(x)=1/x at x=1 find slope of tangent
Answer:
f'(1)=-1
Step-by-step explanation:
You can just find the derivative using the power rule for d/dx (x^-1)=-1x^-2=-1/x^2 then find f'(1)=-1.
I used the limit definition of the derivative to find the slope in the picture attached
Find the area of the shaded portion of the
figure. All angles are right angles.
Dimensions are in inches.
3
8
5
Answer:
22 [in.²]
Step-by-step explanation:
1. the required area can be calculated as
A=A1-A2, where A1- area of big quadrilateral, A2-area of the small quadrilateral;
2. A=5*8-3*6=40-18=22 [in²].
Given a_4=135\;and\;a_6=1215a
4
=135anda
6
=1215, find the explicit rule for the geometric sequence, given that the common ratio is positive.
Answer:
Step-by-step explanation:
let a be first term and r be c.r.
a_{n}=ar^{n-1}
a_{4}=ar^{4-1}=ar^3=135
a_{6}=ar^{6-1}=ar^5=1215a_{4}=1215 \times 135
divide
r^2=1215
r=√1215=35
a_{n}=35 a_{n-1}
If the line segment AB is on a number line, and point A is at 6 and Point B is at 13, then the length of AB is ____.
Answer:
7
Step-by-step explanation:
The length of AB is the distance between A and B which is B-A -> 13-6= 7
12x² + 9x² =
A. 21x²²
B. 21x
C. 22x²
D. 22xA
Answer:
21x²
Step-by-step explanation:
hopefully A is a typo
Answer: A (if it's just a typo) or [tex]21x^{2}[/tex]
Step-by-step explanation:
Since it's the same variable and same exponent, you just need to add the whole numbers and put the same variable and exponent.
12 + 9 = 21
[tex]21x^{2}[/tex]
Which of the numbers below are whole numbers?
A. 546
B. 7317
C. 1/3
D. 47
E. 93.3.
F. 907.68
The volume of the above solid is
A baby weighed only 8.7 ounces at birth. How much lighter was she than an average baby, who weighs about 7 lb 8 ounces?
The baby is 111.3 ounces or 6.96 pounds lighter than an average baby.
What is weight ?Weight is the measurement of Gravitational force on an object.
It can be zero as in the space.
It is given in the question that
A baby weighed only 8.7 ounces at birth
weight of an average baby is 7lb 8 ounces
1 lb = 16 ounce
an average baby weight = 7 * 16 +8 = 120 ounces
The baby weight is just 8.7 ounces
so the baby is 120-8.7 = 111.3 ounces lighter than an average baby.
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Select all radical expressions that are equivalent to...
Answer:
B AND E
Step-by-step explanation:
hope that helpsndbwbe
The scale on a map says 1 inch = 25 miles. If two towns are 3 1/2
apart on the map, what actual distance separates them?
Answer:
87.5 miles
Step-by-step explanation:
Distance between points = 3 1/2 × 25 = 87.5 miles
What linear function can be represented by the set of ordered pairs?
{(−1,−10),(3,2),(5,8),(7,14)}
Enter your answer in the box.
find the value of x
d:6
[tex] \sf{\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Let's solve for x ~
[tex]\qquad \sf \dashrightarrow \: 5x = 3x + 12[/tex]
[ by vertical opposite angle pair ]
[tex]\qquad \sf \dashrightarrow \: 5x - 3x = 12[/tex]
[tex]\qquad \sf \dashrightarrow \: 2x = 12[/tex]
[tex]\qquad \sf \dashrightarrow \: x = 12 \div 2[/tex]
[tex]\qquad \sf \dashrightarrow \: x = 6[/tex]
Therefore, the correct choice is D. 6
A snack bar offers chili and/or cheese on its hot dogs for an additional fee.
The owner kept track of the hot dog orders over the course of a week. His
data are shown in a relative frequency table.
Answer:
the answer is c. the row is chili, the column is no cheese
A 450-pound circus tiger eats 15 pounds of meat per day. How many pounds of meat would you expect a 360-pound tiger to eat per day? A 450 - pound circus tiger eats 15 pounds of meat per day . How many pounds of meat would you expect a 360 - pound tiger to eat per day ?
P(-4, 6), Q(2, 5), R(3, -1), S(-3, 0)
Use slope to determine whether the following is a parallelogram. Explain why or why not.
Find the slope of PQ
Find the slope of QR
Find the slope of RS
Find the slope of SP
Use this information to determine if the quadrilateral a parallelogram.
From the slopes it is confirmed that PQ || RS OR|| SP and therefore it is a parallelogram .
What is a Parallelogram ?A parallelogram is a quadrilateral in which the opposite sides are parallel.
The coordinates of a quadrilateral is given
To prove that it is a parallelogram
PQ || RS
OR|| SP
The slope of a straight line is given by
[tex]\rm m = \dfrac{y_2 -y_1}{x_2- x_1}[/tex]
For PQ
m = (-1/6) = -1/6
For QR
m = -6/1 = -6
For RS
m = 1/(-6) = -1/6
For SP
m = -6
From the slopes it is confirmed that
PQ || RS
OR|| SP
and therefore it is a parallelogram
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What does each decreasing portion of the graph represent? bouncing a ball
Each decreasing portion of the graph indicates the ball has been thrown upward and is traveling toward the ground.
What is a graph?A graph can be defined as a type of chart that is used to graphically represent data on both the horizontal and vertical lines of a Cartesian coordinate i.e x-axis and y-axis.
By critically observing the graph attached in the image below, we can infer and logically deduce that each decreasing portion of the graph indicates the ball has been thrown upward and is traveling toward the ground.
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y=x^2-4x-1 in vertex form
considering the inequality -5x-4y<12 , the shaded area of its graph is described by
The shaded area for the inequality y > -3-(5/4)x is given below.
The inequality is the relation between two expressions showing the relationships such as greater than, lower than, greater than equals to, and lower than equals to.
The given inequality is -5x-4y<12
solving this inequality for y
-5x-4y<12
⇒ -4y<12+5x
⇒y > (12+5x)/(-4)
⇒y > -3-(5/4)x
the line y= -3-(5/4)x has y intersection at (0,-3) and x intersection at (-12/5,0).
The inequality y > -3-(5/4)x show that this will be above the straight line y = -3-(5/4)x as if we put (0,0) in the inequality, the inequality will be 0>-3 which is true that shows that the region contains (0,0).
Therefore the shaded area for the inequality y > -3-(5/4)x is given below.
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The graph of y=x−2−−−−√ is is transformed to become y=x+3−−−−√−2. Which of the following statements best describes the effect this transformation has on the graph of y=x−−√
Answer:
The graph is translated 5 units left and 2 units down
Step-by-step explanation:
transformation =
initially the equation of graph [tex]y=\sqrt{x-2} = f(x)[/tex]
after transformation the equation becomes [tex]y = \sqrt{x+3} -2 = g(x)[/tex]
transformation done [tex]f(x) = \sqrt{x-2}[/tex] → [tex]g(x) = \sqrt{x+3} -2[/tex]
The horizontal shift of graphit depends on the value of h. The horizontal shift is described as:
g(x) = f(x + h) - The graph is shifted to the left h units.
g(x) = f(x - h) - The graph is shifted to the right h units.
if h=0, means that the graph is not shifted to the left or right
the vertical shift of graphThe vertical shift depends on the value of k. The vertical shift is described
as:
g(x) = f(x) +k - The graph is shifted up k units.
g(x) = f(x)- k - The graph is shifted down k units.
so here in the question the graph is shifted 5 units left and then 2 units down
you can see the graph of initial equation and transformed equation below.
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Use the point–slope formula to write an equation of the line that passes through(-5, -16 )and has a slope of m=3. Write the answer in slope–intercept form (if possible).
Answer:
Slope-intercept form: y = 3x - 1
Step-by-step explanation:
We are given that a line passes through the point (-5, -16) and has a slope of 3
We want to write the equation in point-slope form, and simplify it to slope-intercept form (if it is possible to do so).
First, point-slope form is given as [tex]y-y_1=m(x-x_1)[/tex], where m is the slope and [tex](x_1, y_1)[/tex] is a point
We have both the slope and a point, but let's label the values of everything to avoid confusion + mistakes:
m = 3
[tex]x_1=-5\\y_1=-16[/tex]
Now substitute into the formula (remember: we have NEGATIVE numbers, and the formula uses SUBTRACTION):
[tex]y--16=3(x--5)[/tex]
The equation can be simplified
[tex]y+16=3(x+5)[/tex]
We can also write this in slope-intercept form
Start by distributing 3 to both x and 5 (multiply both x and 5 by 3)
[tex]y+16=3x + 15[/tex]
Subtract 16 from both sides
y = 3x - 1
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5+9x+3gg3g3\
1 b
Johann is 60 and Chad is 34 years younger. How many years
ago was Johann three times as old as Chad?
Answer: 6 years ago
Step-by-step explanation: Since Chad is 34 years younger than Johann, Chad is currently 26. 60 divided by 3 (since the question says three times) is 20, and 26 (Chad’s current age) minus 20 is 6. So, 6 years is the correct answer.
Answer:
21 years
Step-by-step explanation:
Let x represent the number of years since Johann was 3 times Chad's age. The given relation can be written as the equation ...
(60 -x) = 3(34 -x)
__
Solving this, we have ...
60 -x = 102 -3x . . . . . eliminate parentheses
2x = 42 . . . . . . . . . add 3x-60
x = 21 . . . . . . . . .divide by 2
21 years ago, Johann was 3 times as old as Chad.
_____
Additional comments
Their ages then were 39 and 13.
If you recognize the age difference is 60-34 = 26 years, and that J will be 3×C when C's age is half that difference, then you know C was 13 at that time. That was 34-13 = 21 years ago.