The dilation transformation is determined by the scale factor and the centre of dilation. The image stretches when the scaling factor exceeds 1. When the scale factor is between 0 and 1, the image gets smaller.
What is scale factor of dilation?Ratio of scale (Dilation) How much larger or smaller the image will be depends on the scale factor in a mathematical object's dilatation (compared to the original object). When the scaling factor's absolute value exceeds 1, an expansion takes place.The x and y values of each point will be multiplied by 2 to enlarge the graphic by a factor of 2. To identify the new triangle, I plotted each new point. The x-value of each point will be multiplied by 2 to enlarge the image by a factor of two.A scale factor is defined as the ratio of an original object's scale to that of a new, smaller version of the original object.To learn more about dilation refer to:
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The dilation transformation is determined by the scale factor and the center of dilation. The image stretches when the scaling factor exceeds 1. When the scale factor is between 0 and 1, the image gets smaller.
What is scale factor of dilation?Ratio of scale (Dilation) How much larger or smaller the image will be depends on the scale factor in a mathematical object's dilatation (compared to the original object). When the scaling factor's absolute value exceeds 1, an expansion takes place.
The x and y values of each point will be multiplied by 2 to enlarge the graphic by a factor of 2. To identify the new triangle, I plotted each new point. The x-value of each point will be multiplied by 2 to enlarge the image by a factor of two.
A scale factor is defined as the ratio of an original object's scale to that of a new, smaller version of the original object.
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The Drury Lane Theater has 10 seats in the first row and 38 rows in all. Each successive row contains one additional seat. How many seats are in the theater?
Anyone is here
Answer:
1083 seats
Step-by-step explanation:
this is an arithmetic sequence :
an = an-1 + c
in our case
a1 = 10 (10 seats in the first row)
a2 = a1 + 1 = 10 + 1 = 11
a3 = a2 + 1 = a1 + 1 + 1 = 10 + 1 + 1 = 12
an = an-1 + 1 = a1 + (n-1)×1 = a1 + n - 1 = 10 + n - 1 = 9 + n
a38 = a1 + 37 = 10 + 37 = 47 seats.
altogether this means the sum of the arithmetic sequence of a1 to a38.
the general sum of an arithmetic sequence from a1 to an is
n×(a1 + an)/2
in our case
38×(10 + 47)/2 = 19×57 = 1083 seats
find the missing length indicated 144 and 60
The missing length in the right triangle as given in the task content is; 156.
What is the missing length indicated?It follows from the complete question that the triangle given is a right triangle and the missing length (longest side) can be evaluated by means of the Pythagoras theorem as follows;
x² = 144² + 60²
x² = 20736 + 3600
x² = 24,336
x = √24336
x = 156.
Remarks: The complete question involves a right triangle and the missing length is the longest side.
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. Find the number of possible call letters of a radio station (8 letters long) if the first letter must be a W or a K and no letter is repeated\.\*
the number of possible call letters of a radio station (8 letters long) if the first letter must be a W or a K and no letter is repeated is 4, 845, 456, 0000 call letter
How to determine the numberFrom the given information, we need to find the number which is
call letters to a radio station is 7 letters longthe first letter must be a W or a K No letters are repeated.First letter have only two possibility.
But we know that there are a total of 26 letters in alphabet.
First letter must be W or K. So remaining 25 letters.
The Remaining 7 letters can be any alphabet (25 letters)
We also know that np letters are repeated.
So, the second letters have 25 possibility and the third letter have 24 possibility and so one;
Thus, we have
= 2 × 25 × 24 × 23 × 22 × 21 × 20 × 19
Multiply through
= 4, 845, 456, 0000 call letter
Thus, the number of possible call letters of a radio station (8 letters long) if the first letter must be a W or a K and no letter is repeated is 4, 845, 456, 0000 call letter
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Someone please help meeeeeee
Answer:
a ≈ 16.5 cm , b ≈ 23.8 cm
Step-by-step explanation:
using the Law of Sines in Δ ABC
[tex]\frac{a}{sinA}[/tex] = [tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex]
we require to calculate ∠ C
∠ C = 180° - (42 + 75)° = 180° - 117° = 63°
Then to find a
[tex]\frac{a}{sinA}[/tex] = [tex]\frac{c}{sinC}[/tex] ( substitute values )
[tex]\frac{a}{sin42}[/tex] = [tex]\frac{22}{sin63}[/tex] ( cross- multiply )
a × sin63° = 22 × sin42° ( divide both sides by sin63° )
a = [tex]\frac{22sin42}{sin63}[/tex] ≈ 16.5 cm ( to the nearest tenth )
similarly to find b
[tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex] ( substitute values )
[tex]\frac{b}{sin75}[/tex] = [tex]\frac{22}{sin63}[/tex] ( cross- multiply )
b × sin63° = 22 × sin75° ( divide both sides by sin63° )
b = [tex]\frac{22sin75}{sin63}[/tex] ≈ 23.8 cm ( to the nearest tenth )
Answer:
Step-by-step explanation:
Sine rule of Law of sine:[tex]\sf \boxed{\bf\dfrac{a}{Sin \ A}=\dfrac{b}{Sin \ B}=\dfrac{c}{Sin \ C}}[/tex]
Side 'a' faces ∠A.
Side 'b' faces ∠B.
Side 'c' faces ∠C.
We have to find ∠C using angle sum property of triangle.
∠C + 75 + 42 = 180
∠C +117 = 180
∠C = 180 - 117
∠C = 63°
[tex]\sf \dfrac{a}{Sin \ 42}= \dfrac{22}{Sin \ 63}\\\\ \dfrac{a}{0.67}=\dfrac{22}{0.89}\\\\[/tex]
[tex]\sf a = \dfrac{22}{0.89}*0.67\\\\ \boxed{a = 16.56 \ cm }[/tex]
[tex]\sf \dfrac{b}{Sin \ B} = \dfrac{c}{Sin \ C}\\\\ \dfrac{b}{Sin \ 75}=\dfrac{22}{Sin 63}\\\\ \dfrac{b}{0.97} =\dfrac{22}{0.89}\\\\[/tex]
[tex]\sf b = \dfrac{22}{0.89}*0.97\\\\ \boxed{b =23.98 \ cm }[/tex]
If 3 square feet of fabric costs $3.75. what would 7 square feet cost?
Answer:
8.75
Step-by-step explanation:
3.75/3=1.25
1.25=1 square feet
1.25(7)=8.75
For question 1-8 calculate the surface area. If necessary round to the nearest hundredth
Surface area = 400 unit square
Surface area = 1100 unit square
Surface area = 1100 unit square
Surface area = 2400 unit square
How to determine the surface area
The first figure is a cuboid;
Surface area of a cuboid = 2lw+2lh+2hw
L = 9
W = 11
h = 4
Surface area = 2( 9 × 11) + 2(9 × 4) + 2( 4× 11)
Surface area = 2(99) + 2(36) + 2(44)
Surface area = 198 + 72 + 88
Surface area = 400 unit square
Surface area of a triangular prism = Area of top + area of base
Area of the top = 1/ 2 b × h
Area = 1/2 × 4 × 8
Area = 16 unit square
Area of base = length × width
Area = 4× 4
Area = 16 unit square
Surface area = 16 + 16
Surface area = 1100 unit square
Surface area of a cylinder = 2πrh + 2πr² = 2 ( 3. 412 × 7. 5 × 15) + 2 ( 3. 141× 7. 5 × 7.5)
Surface area = 706. 95 + 353. 48
Surface area = 1100 unit square
Surface area of a cone =πr(r+ √h² + r2)
Surface area = [tex]3. 142 * 15 ( 15 + \sqrt{10^2 +34^2 }[/tex]
Surface area = [tex]47. 13 * 50. 44[/tex]
Surface area = 2400 unit square
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Select the correct answer.
Which statement correctly compares functions f and g?
Answer:
C
Step-by-step explanation:
Exponential functions have 2 things in common
-They increase infinitely once they approach a certain point
-They don't decrease anymore once they approach a certain point
Elias and niko are polishing the silver at the heritage museum. elias could polish all the silver himself in 40 minutes. that task would take niko 50 minutes to complete alone. which table is filled in correctly and could be used to determine how long it would take if elias and niko polished the silver together? a table showing rate in part per minute, time in minutes, and part of silver polished. the first row shows elias, and has startfraction 1 over 40 endfraction, t, and startfraction 1 over 40 endfraction t. the second row shows niko and has, startfraction 1 over 50 endfraction, t, and startfraction 1 over 50 endfraction t. a table showing rate in part per minute, time in minutes, and part of silver polished. the first row shows elias, and has r, startfraction 1 over 40 endfraction, and startfraction 1 over 40 endfraction r. the second row shows niko and has, r, startfraction 1 over 50 endfraction, and startfraction 1 over 50 endfraction r. a table showing rate in part per minute, time in minutes, and part of silver polished. the first row shows elias, and has 40, t, and 40 t. the second row shows niko and has, 50, t, and 50 t . a table showing rate in part per minute, time in minutes, and part of silver polished. the first row shows elias, and has r, 40, and 40 r. the second row shows niko and has, r 50, and 50 r .
The first table is filled correctly and can be used to determine how long it takes for Elias and Niko to polish the silver.
How to interpret Mathematical Tables?The first table is filled correctly and can be used to determine how long it takes for Elias and Niko to polish the silver.
The rate is the unit rate.
Now, since Elias polishes all of the silver himself in 40 minutes, his rate is 1/40.
Similarly, Niko’s rate would be 1/50.
Their time is the same because they are working at the same time, and as such we can write that as t. Then, the part of the silver polished would be the rate * time, or:
Thus;
Part polished by Elias: ¹/₄₀ * t
Part Polished by Niko: ¹/₅₀ * t
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A square piece of paper measures 20
centimeters on each side. Four equally-
sized circles are going to be cut out from
the paper. What is the largest possible
area of ONE of the circles?
The largest possible area of one of the circles would be 314. 2 cm²
Area of a circleIt is important to note that the formula for finding the area of a circle is given as;
Area = [tex]\pi r^2[/tex]
From the information given, we have the sides of the square to have a value of 20cm
Also note that the measure of the sides of the square is the size of the diameter of the circle
But we need to find the radius
radius = diameter/2
radius = 20/ 2 = 10cm
Substitute value of radius into the formula for area
Area = 3. 142 × 10^2
Area = 3. 142 × 10 × 10
Area = 314. 2 cm²
The area of the circle is 314. 2cm^2
Thus, the largest possible area of one of the circles would be 314. 2 cm²
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which expression uses the associtive property to make it easier to evaluate 8 (1/4 x 3/5)
The value of the expression 8 (1/4 * 3/5) is 6/5
How to evaluate the expression?The expression is given as:
8 (1/4 * 3/5)
The associative property states that:
a * (b * c) = (a * b) * c
Using the above expression, we have:
8 (1/4 * 3/5) = 8 * 1/4 * 3/5
Evaluate the product
8 (1/4 * 3/5) = 2 * 3/5
This gives
8 (1/4 * 3/5) = 6/5
Hence, the value of the expression 8 (1/4 * 3/5) is 6/5
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Help all of these are confusing me
Answer:
see explanation
Step-by-step explanation:
the equation of a parabola in vertex form is
y = a(x - h)² + k
where (h, k ) are the coordinates of the vertex and a is a multiplier
here (h, k ) = (1, - 4 ) and a = 1 , then
y = (x - 1)² - 4 → b
------------------------------------
given 3 sides of a triangle then an angle may be found using the cosine law. → b
if the 3 sides are a, b, c then
cosA = [tex]\frac{b^2+c^2-a^2}{2bc}[/tex] ← allowing ∠ A to be found
--------------------------------------
cosC = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{BC}{AC}[/tex] = [tex]\frac{4}{5}[/tex] → a
-----------------------------------------
since the triangles are similar then the ratios of corresponding sides are in proportion, that is
[tex]\frac{AB}{DE}[/tex] = [tex]\frac{BC}{EF}[/tex] ( substitute values )
[tex]\frac{x}{6}[/tex] = [tex]\frac{10}{4}[/tex] ( cross- multiply )
4x = 6 × 10 = 60 ( divide both sides by 4 )
x = 15 → b
Answer:
bbabStep-by-step explanation:
There are a few algebraic, geometric, and trig relations you are expected to remember. These come into play in this set of questions.
vertex form for equation of a parabola: y = a(x -h)² +k, has vertex at (h, k)Sine Law relates triangle sides and their opposite angles: a/sin(A) = b/sin(B)Cosine Law relates triangle sides and the angle between two of them: c² = a² +b² -2ab·cos(C)SOH CAH TOA reminds you of trig relations in a right trianglerelationships of corresponding sides and angles in congruent and similar triangles: angles are congruent; sides are congruent or proportional.When solving any problem, the first step is to understand what is being asked. The second step is to identify the relevant information and relationships that can help you answer.
1)You are asked for the equation of a parabola with a given vertex. The vertex form equation will be useful. We can assume a scale factor ('a') of 1.
For vertex (h, k) = (1, -4) and a=1, the vertex form equation is ...
y = a(x -h)² +k
y = 1(x -1)² +(-4)
y = (x -1)² -4
2)You are given 3 sides and want to find an angle. The useful relation in this case is the Cosine Law. (If you wanted to use the Sine Law, you would already need to know an angle.)
3)The mnemonic SOA CAH TOA reminds you that the cosine relation is ...
Cos = Adjacent/Hypotenuse
The side adjacent to angle C is marked 4; the hypotenuse is marked 5. The desired ratio is ...
cos(C) = 4/5
4)The measure x is also the measure of side AB. The similarity statement lists those letters as the first two. It also lists the letters DE as the first two. The other given side in ΔABC is BC, corresponding to side EF in the smaller triangle. Corresponding sides are proportional, so we have ...
AB/DE = BC/EF
x/6 = 10/4
We can find the value of x by multiplying this equation by 6:
x = 6(10/4) = 60/4
x = 15
Please note that BC is the shortest side in ΔABC. This means x > 10. There is only one such answer choice. (No math necessary.)
See photo to answer!
The length of AB of the triangle is 13 units.
How to find the side of a triangle?The side AB of the triangle can be found using cosine law,
Therefore,
c² = a² + b² - 2ab cos C
c² = 7² + 8² - 2 × 7 × 8 cos 120
c² = 49 + 64 - 112 cos 120
c²= 113 - (-56)
c² = 113 + 56
c² = 169
c = √169
c = 13 units
Therefore, the value of AB is 13 units
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Please help with these 3 questions
Answer:
dbbStep-by-step explanation:
Trigonometry is a branch of mathematics that relates the measures of sides and angles in a triangle. The various trig functions of an angle are defined in terms of the sides of a right triangle containing that angle. Consequently, the Pythagorean theorem can be used to relate certain trig functions to each other, and to obtain formulas for the sum and difference of angles.
1. Solving trianglesThe Pythagorean theorem relates side measures in right triangles. For relating side measures to each other or to angles in non-right triangles, one or both of the Law of Sines and the Law of Cosines must be used.
both b) and c)
2. Finding an angleThe Law of Sines relates side lengths and the sines of angles. Knowing three side lengths tells you nothing about the angles except the ratio of their sines.
The Law of Cosines relates three side lengths and one angle. Knowing the three side lengths, the angle measure can be found.
There is no Tangent Law.
3. Use of trigonometryTrigonometry is a calculation tool, not a measuring tool. As such, it allows calculation of angles and sides (of a triangle) that cannot be physically measured.
Paintings benefit greatly from rules of proportion and perspective. In general, trigonometry is not directly involved. (Computer-generated art may implement these rules of perspective using trigonometry. Paintings, in general, make no direct use of trigonometry.)
x =_______ y =_______ 12. Find the values of x and y.
Answer:
Below in bold.
Step-by-step explanation:
Opposite angles of a quadrilateral i a circle are supplementary so
x = 180 - 112 = 68 degrees.
y = 180 - 81 = 99 degrees.
Assume that y varies directly with x. If y = 24 when
x = 6, find y when x = -4.
y=
Check
Step-by-step explanation:
that means nothing else than
y = k×x
24 = k×6
k = 24/6 = 4
y = k×-4 = 4×-4 = -16
Y=16 x 10^8k , where k is an integer. find an expression, in terms of k , for y^5/4
The requried simplified expression for [tex]y^{5/4}[/tex] is given by [tex]y^{5/4}= 32 * 10^{10k}[/tex]
To find an expression for [tex]y^{5/4}[/tex] in terms of k, we'll substitute the given value of y into the expression and then apply the exponent (5/4).
Given: [tex]y = 16 *10^{8k}[/tex]
Now, let's calculate [tex]y^{5/4}[/tex]:
[tex]y^{5/4} = (16 *10^{8k})^{5/4}[/tex]
To apply the exponent (5/4), we raise each factor to the power of (5/4):
[tex]y^{5/4}= 16^{5/4} * (10^{8k})^{5/4}[/tex]
Since [tex](10^8)^k[/tex] is [tex]10^{(8k)}[/tex], we have:
[tex]y^{5/4}= 16^{5/4}* 10^{8k * 5/4)[/tex]
[tex]y = 16*10^{8k}[/tex]
[tex]y^{5/4}= 2^5 * 10^{10k}[/tex]
Finally, we can write the expression in terms of k:
[tex]y^{5/4}= 32 * 10^{10k}[/tex]
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What divided by 4 equals 9/4
( 9/4= Divided by 4 )
Answer
16/9
Step-by-step explanation:
I don't know if this is right or not and I'm sry if its wrong
but 16/9 divided by 4 = 9/4
find the value of x for which the fraction 4x +9/7x
Answer:
x = 3
Step-by-step explanation:
Here ,
4x + 9 =7x
--> 4x - 7x = -9
--> -3x = -9
--> ≠ x = -9/3
--> x = 3
HELP, WILL GIVE BRAINLEST..
Factor the GCF: -6x³y + 9x²y2 - 12xy³ (5 points)
O 3xy(-2x² + 3xy - 4y²)
O-3xy(2x² + 3xy - 4y²)
O-3xy(2x2-3xy + 4y²)
O-3(2x³y - 3x²y² + 4xy³)
Answer:
[tex]\sf -3xy\left(2x^2-3xy+4y^2\right)[/tex]
Step-by-step explanation:
[tex]\sf -6x^3y+9x^2y^2-12xy^3[/tex]
To factor the GCF of 6x³y + 9x²y2 - 12xy³ let's apply the exponent rule:-
[tex]\boxed{\sf a^{b+c}=a^ba^c}[/tex]
[tex]\boxed{\sf x^3y=xx^2y,\:x^2y^2=xxyy,\:xy^3=xyy^2}[/tex]
[tex]\sf -6xx^2y+9xxyy-12xyy^2[/tex]
Rewrite,
-6 as 2 * 39 as 3 * 3-12 as 4 * 3[tex]\sf 2\cdot \:3xx^2y+3\cdot \:3xxyy+4\cdot \:3xyy^2[/tex]
Now, factor out the common term [tex]\sf -3xy[/tex]:-
[tex]\sf -3xy\left(2x^2-3xy+4y^2\right)[/tex]__________________________
Recently, the stock market took big swings up and down. A survey of 969 adult investors asked how often they tracked their portfolio. The table shows the investor responses. What is the probability that an adult investor tracks his or her portfolio daily
A survey of 969 adult investors asked how often they tracked their portfolio. The probability that an adult investor tracks his or her portfolio daily is 0.238.
From the table it is clear that the number of responses daily by the adult investors is 231.
What is probability?Probability is the ration of the number of possible value to the total value.
Here the number of daily response by the adult investors = 231
The total adult investors = 969
Hence, the The probability that an adult investor tracks his or her portfolio daily = 231/969 = 0.238.
Therefore, in the total number of 969 adult investors, the probability that the investors response daily is 0.238.
Disclaimer: The question is incomplete. The following is the correct question.
Question: Recently, the stock market took big swings up and down. A survey of 969 adult investors asked how often they tracked their portfolio. The table shows the investor responses. What is the probability that an adult investor tracks his or her portfolio daily
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If a fair coin is tossed four times, what is the probability of it landing heads up at least three times
The probability of it landing heads up at least three times is 1/4.
According to the statement
We have a given that the coin is tossed 4 times And we have to find the probability of it landing heads up at least three times.
So, we know that the
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1.
So, According to the statement
Coin is tossed 4 times and we have to probability of find heads three times it means n = 3.
To get 3 heads, means that one gets only one tail. This tail can be either the 1st coin, the 2nd coin, the 3rd, or the 4th coin.
Thus there are only 4 outcomes which have three heads.
The probability is 4/16 = 1/4.
So, The probability of it landing heads up at least three times is 1/4.
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Which graph models the pH of this solution?
The graph that models the pH of this solution is graph A.
What is a graph?It should be noted that a graph is a diagram such as a series of one or more points, lines, line segments, curves, or areas which represents the variation of a variable in comparison with that of one or more other variables.
The pH is a measure of how acidic or basic water is. It should be noted that the range goes from 0 - 14, with 7 being neutral. The pHs of less than 7 indicate acidity, while a pH of greater than 7 indicates a base. The pH is really a measure of the relative amount of free hydrogen and hydroxyl ions that are in the water.
In this case, x represents the concentration of the hydrogen ions. The first graph illustrates this.
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Complete question:
The ph of a particular solution is given by pH=-log(x-2) where x represents the concentration of the hydrogen ions in the solution in moles per liter. Which graph models the ph of this solution?
Question 6
10 pts
Which is a counterexample for the following biconditional: "A figure is a quadrilateral if and only if it
is a polygon"?
Answer:
Trapezium, Rhombus, Kite, etc.
Step-by-step explanation:
A four - sided figure.
The name of a counterexample for the following biconditional: "A figure is a quadrilateral if and only if it is a polygon" is Triangle.
Used the concept of the polygon that states,
In geometry, a polygon can be defined as a flat or plane, two-dimensional closed shape bounded by straight sides.
Given the condition is,
"A figure is a quadrilateral if and only if it is a polygon"
Now, we know that;
If a polygon is a quadrilateral, then it has four sides, and if a polygon has four sides, then it is a quadrilateral.
Hence, Triangle is a counterexample for the biconditional "A figure is a quadrilateral if and only if it is a polygon."
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The perimeter of a square is 40 inches.
What is the area, in square inches, of the square?
The area of the square is 100 inches square
What is perimeter of a square?The perimeter of a square is calculated using the formula;
Perimeter = 4 a
Where
a is the side
It is also important to know that the area of a square is calculated using the formula;
Area = a²
Where a represents the side of the square
Let's find the side by substituting the value of the perimeter in the formula, we have
40 = 4a
a = 40/4
a = 10 inches
Then , let's find the area by substituting the value of the side
Area = 10 ²
Area = 10 × 10
Area = 100 inches square
Thus, the area of the square is 100 inches square
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An aircraft is timetabled to travel from a to b. due to bad weather it flies from a to c then from c to b, where ab and cb make angle of 27 degree and 66 degree respectively with ab if [ab]= 220km calculate [ ab]
Answer:240.5
Step-by-step explanation:
Angle C = 87°(calculated)
AB=C
AC=220=b
Sine laws c\sin87=220\66
c=240.5
Sutopa has 3/4 of the money Maneet has. Maneet has $18 less than Kim. Together, they have $286.40. How much money does each of them have?
Solving a system of equations we can see:
Sutopa has $73.20Maneet has $97.60Kim has $115.60How much money each of them has?
First, let's define the variables:
S = money that Sutopa has.M = money that Maneet has.K = money that Kim has.We can write the system of equations:
S = (3/4)*M
M = K - $18
S + M + K = $286.40
First, we can rewrite the second equation to get:
K = M + $18
Now we can replace the first and second equations into the third one:
(3/4)*M + M + (M + $18) = $286.40
Now we can solve this for M:
M*(1 + 1 + 3/4) = $286.40 - $18
M = $268.40*(4/11) = $97.60
Now we can find the other two values:
K = M + $18 = $97.60 + $18 = $115.60
S = (3/4)*M = (3/4)*$97.60 = $73.20
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Anthony is rowing a boat upstream. The following equation models his speed: f(x) = 3x2 − 6x − 13, where x is the velocity of the boat relative to land. What is the domain of the function?
You work at a pioneer historical site. On this site you have handcarts. One cart has a handle that connects to the center of the wheel. We have to maintain the handle of the cart at an angle of no more than 20° with the ground so the contents do not spill out. The distance from where the handle rests on the ground to the point where the wheel is sitting on the ground is 45 inches. The distance of the center of the wheel to the end of the handle is approximately 48 inches.
a. Identify the parts of the handcart wheel that would represent congruent chords and congruent central angles. Explain why.
b. Find the radius of the wheel.
c. If the measure of the arc from to around the outside of the wheel were changed to 72°, what is the new angle the handle makes with the ground? Will the contents remain in the handcart at that angle? Will the handle rest on the ground?
d. If a pioneer pulling the handcart held the handle at a height of 48 inches off the ground, would the contents of the cart spill out the back? How high can the pioneer lift the handle off the ground before the contents started spilling out?
Answer:
a) see below
b) radius = 16.4 in (1 d.p.)
c) 18°. Yes contents will remain. No, handle will not rest on the ground.
d) Yes contents would spill. Max height of handle = 32.8 in (1 d.p.)
Step-by-step explanation:
Part a
A chord is a line segment with endpoints on the circumference of the circle.
The diameter is a chord that passes through the center of a circle.
Therefore, the spokes passing through the center of the wheel are congruent chords.
The spokes on the wheel represent the radii of the circle. Spokes on a wheel are usually evenly spaced, therefore the congruent central angles are the angles formed when two spokes meet at the center of the wheel.
Part b
The tangent of a circle is always perpendicular to the radius.
The tangent to the wheel touches the wheel at point B on the diagram. The radius is at a right angle to this tangent. Therefore, we can model this as a right triangle and use the tan trigonometric ratio to calculate the radius of the wheel (see attached diagram 1).
[tex]\sf \tan(\theta)=\dfrac{O}{A}[/tex]
where:
[tex]\theta[/tex] is the angleO is the side opposite the angleA is the side adjacent the angleGiven:
[tex]\theta[/tex] = 20°O = radius (r)A = 45 inSubstituting the given values into the tan trig ratio:
[tex]\implies \sf \tan(20^{\circ})=\dfrac{r}{45}[/tex]
[tex]\implies \sf r=45\tan(20^{\circ})[/tex]
[tex]\implies \sf r=16.37866054...[/tex]
Therefore, the radius is 16.4 in (1 d.p.).
Part c
The measure of an angle formed by a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs.
If the measure of the arc AB was changed to 72°, then the other intercepted arc would be 180° - 72° = 108° (since AC is the diameter).
[tex]\implies \sf new\: angle=\dfrac{108^{\circ}-72^{\circ}}{2}=18^{\circ}[/tex]
As the handle of the cart needs to be no more than 20° with the ground for the contents not to spill out, the contents will remain in the handcart at an angle of 18°.
The handle will not rest of the ground (see attached diagram 2).
Part d
This can be modeled as a right triangle (see diagram 3), with:
height = (48 - r) inhypotenuse ≈ 48 inUse the sin trig ratio to find the angle the handle makes with the horizontal:
[tex]\implies \sf \sin (\theta)=\dfrac{O}{H}[/tex]
[tex]\implies \sf \sin (\theta)=\dfrac{48-r}{48}[/tex]
[tex]\implies \sf \sin (\theta)=\dfrac{48-45\tan(20^{\circ})}{48}[/tex]
[tex]\implies \theta = 41.2^{\circ}\:\sf(1\:d.p.)[/tex]
As 41.2° > 20° the contents will spill out the back.
To find the maximum height of the handle from the ground before the contents start spilling out, find the height from center of the wheel (setting the angle to its maximum of 20°):
[tex]\implies \sin(20^{\circ})=\dfrac{h}{48}[/tex]
[tex]\implies h=48\sin(20^{\circ})[/tex]
Then add it to the radius:
[tex]\implies \sf max\:height=48\sin(20^{\circ})+45\tan(20^{\circ})=32.8\:in\:(1\:d.p.)[/tex]
(see diagram 4)
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Circle Theorem vocabulary
Secant: a straight line that intersects a circle at two points.
Arc: the curve between two points on the circumference of a circle
Intercepted arc: the curve between the two points where two chords or line segments (that meet at one point on the other side of the circle) intercept the circumference of a circle.
Tangent: a straight line that touches a circle at only one point.
A dog gave birth to 9 puppies, of which 2 are brindle. What is the ratio of brindle puppies to non-brindled puppies?
Answer:
2:7
Step-by-step explanation:
If 2 of the puppies are brindle that leaves 7 that are not out of the nine making the ratio 2:7.
What is the remainder when f(x) = 2x^3 – 12x^2 + 11x + 2 is divided by x – 5? Show your work.
Answer:
7
Step-by-step explanation:
given a polynomial divide by (x - a) then the remainder is f(a)
here f(x) is divided by (x - 5) , so remainder is
f(5) = 2(5)³ - 12(5)² + 11(5) + 2
= 2(125) - 12(25) + 55 + 2
= 250 - 300 + 57
= - 50 + 57
= 7