Answer:
300 gallons
Step-by-step explanation:
A saline solution is a solution of salt in water.
The percent is the percent of salt in the total amount of solution.
A 40% saline solution has 40% of salt out of the total volume.
Let the amount of pure water needed = x.
The amount of existing 40% solution is 500 gal.
Let the total amount of 25% saline solution made = y.
Equation of volumes of solutions:
500 + x = y
y = x + 500 Eq. 1
Equation of volume of salt:
500 gal of 40% saline solution has 0.4 × 500 gal = 200 gal of salt
Pure water has 0% salt.
The final product is a solution that is 25% saline. Its volume is y.
The volume of salt in the y gallons of 25% saline solution is 25% of y = 0.25y
The equation of salt content is:
200 + 0 = 0.25y
y = 800 Eq. 2
Eq. 1 and Eq. 2 form a system of equations.
y = x + 500
y = 800
Substitute 800 for y in Eq. 1.
y = x + 500
800 = x + 500
x = 300
Answer: 300 gallons of pure water
Which of the following is an equivalent form of the equation of the graph shown in the xyxyx, y-plane above, from which the coordinates of vertex AAA can be identified as constants in the equation
The equivalent form of the equation y=[tex]x^{2} -2x-15[/tex]given is y=(x+3)(x-5).
Given an equation y=[tex]x^{2}[/tex]-2x-15 and we are required to find the equivalent form of the equation.
Equation is like a relationship between two or more variables that are expressed in equal to form. Equation of two variables look like ax+by=c. It may be linear equation, quadratic equation, cubic equation or any other equation depending on the powers of the variable.
To find the equivalent equations we are required to form factors of the equation. Equivalent equation are those equations which when solved gives the same solution as the equation when solved gives.
y=[tex]x^{2}[/tex]-2x-15
y=[tex]x^{2}[/tex]-5x+3x-15
y=x(x-5)+3(x-5)
y=(x+3)(x-5)
Hence the equivalent form of the equation y= [tex]x^{2}[/tex]-2x-15 given is
y=(x+3)(x-5).
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Question is incomplete as question should includes the equation
y= [tex]x^{2}[/tex]-2x-15.
The quadratic parent function has been reflected down, stretched vertically by a factor of 1/3
1a) f(x) = -1/3(x + 12)² + 9
1b) f(x) = -1/3x² - 8x - 39
Lets simplify it,
Expand by FOIL (First Outside Inside Last)
Standard Form: ax² + bx + c = 0
Transformations Graph: f(x) = a(bx - c)² + d
Reflected down and vertically stretched by 1/3: a = -1/3
Shifted vertically by 9 units: d = 9
Shifted horizontally by -12 units: c = -12
Vertex Form:
f(x) = a(bx - c)² + d
f(x) = -1/3(x + 12)² + 9
Standard Form:
f(x) = -1/3(x + 12)² + 9
f(x) = -1/3(x² + 24x + 144) + 9
f(x) = -1/3x² - 8x - 48 + 9
f(x) = -1/3x² - 8x - 39
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A population of a particular yeast cell develops with a constant relative growth rate of 0.4311 per hour. The initial population consists of 3.9 million cells. Find the population size (in millions of cells) after 5 hours. (Round your answer to one decimal place.)
A population of a particular yeast cell develops with a constant relative growth rate of [tex]0.4311[/tex] per hour. The initial population consists of [tex]3.9[/tex]million cells. Find the population size (in millions of cells) after 5 hours. Population size after [tex]5hrs[/tex] is [tex]33.6million[/tex].
How can we find the population size ?
The projection of population growth in yeast is given by
[tex]N=N_{0} e^{rt}[/tex]
Where [tex]N_{0}[/tex]=initial population which is [tex]3.9million[/tex]
[tex]r[/tex]=intrinsic rate of natural increases which is [tex]0.4311million per hour[/tex]
N is population size
Substitute the values
[tex]N=N_{0} e^{rt}\\N=3.9(e^{0.4311*5} )\\\\N=33.6 million[/tex]
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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.
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?
. 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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Which triangles are similar to ABC? Explain.
Answer: △JKL & △ MNP
Step-by-step explanation:
△JKL is similar because:
5 x 1.6 = 8
2.5 x 1.6 = 4
3.75 x 1.6 = 6
△MNP is also similar because:
4 x 2 = 8
2 x 2 = 4
3 x 2 = 6
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
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]
A sphere and a cylinder have the same radius and height. The volume of the cylinder is . Amie found the volume of the sphere. Her work is shown below. What is Amies error
Amie's error while measuring the volume of a sphere is that Amie should have multiplied 54 by 2/3. Thus, the first option is the right choice.
In the question, we are given that a sphere and a cylinder have the same radius and height.
We assume the radius of the sphere to be r, and its height to be h.
Now, the height of a sphere is its diameter, which is twice the radius.
Thus, the height of the sphere, h = 2r
Given that the sphere and the cylinder have the same radius and height, the radius of the cylinder is r, and its height is 2r.
The volume of a sphere is given by the formula, V = (4/3)πr³, where V is its volume, and r is its radius.
Thus, the volume of the given sphere using the formula is (4/3)πr³.
The volume of a cylinder is given by the formula, V = πr²h, where V is its volume, r is its radius, and h is its height.
Thus, the volume of the given cylinder using the formula is πr²(2r) = 2πr³.
Now, to compare the two volumes we take their ratios, as
Volume of the sphere/Volume of the cylinder
= {(4/3)πr³}/{2πr³}
= 2/3.
Thus, the volume of the sphere/the volume of the cylinder = 2/3,
or, the volume of the sphere = (2/3)*the volume of the cylinder.
Given the volume of the cylinder to be 54 m³, Amie should have multiplied 54 by 2/3 instead of adding the two.
Thus, Amie's error while measuring the volume of a sphere is that Amie should have multiplied 54 by 2/3. Thus, the first option is the right choice.
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For the complete question, refer to the attachment.
2+8+32+128…..,n=9
Evaluate each geometric series described
The value of 2+8+32+128…..,n=9 is 174762
How to evaluate the series?The series is given as:
2+8+32+128…..,n=9
Start by calculating the common ratio (r)
r = 8/2
r = 4
The sum of the series is then calculated as:
[tex]S_n = \frac{a *(r^n - 1)}{r -1}[/tex]
This gives
[tex]S_9 = \frac{2 *(4^9 - 1)}{4 -1}[/tex]
Evaluate the difference
[tex]S_9 = \frac{2 *( 262143)}{3}[/tex]
Evaluate the quotient
[tex]S_9 = 174762[/tex]
Hence, the value of 2+8+32+128…..,n=9 is 174762
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Study the illustration. Write the ratio
of flowers to bees. Complete the sentence.
For every 5 flowers there are _______
bees.
Answer: 6 bees
Step-by-step explanation:
The picture has 12 bees and 10 flowers. Thus, the ratio of flowers to bees is 10 to 12, which simplifies to 5 to 6.
Therefore, for every 5 flowers, there are 6 bees.
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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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?
Let k be a positive integer. In how many ways can one select three distinct numbers from the set {1,2,..., 3k} such that their sum is divisible by 3
Reduce the numbers in the list modulo 3 to get the set
{1, 2, 0, 1, 2, 0, …, 1, 2, 0}
containing [tex]k[/tex] copies each of 1, 2, and 0.
Take any 3 elements from the list. Their sum is divisible by 3 if those elements' residues also sum to 3 ≡ 0 (mod 3). To get a sum of 0, we must make one of the following choices:
3 elements each with the same residue, so0 + 0 + 0 ≡ 0 (mod 3)
1 + 1 + 1 ≡ 3 ≡ 0 (mod 3)
2 + 2 + 2 ≡ 6 ≡ 0 (mod 3)
1 element each with different residues, so0 + 1 + 2 ≡ 3 ≡ 0 (mod 3)
There are
[tex]\dbinom k3 \dbinom k0 \dbinom k0 = \dfrac{k(k-1)(k-2)}6[/tex]
ways of choosing 3 elements with a given residue and 0 elements with any other residue, hence
[tex]3\dbinom k3\dbinom k0\dbinom k0 = \dfrac{k(k-1)(k-2)}2[/tex]
ways of choosing any 3 elements with the same residue, and there are
[tex]\dbinom k1 \dbinom k1 \dbinom k1 = k^3[/tex]
ways of choosing any 3 elements with distinct residues.
So, the total number of ways of making the selection is
[tex]3\dbinom k3\dbinom k0^2 + \dbinom k1^3 = \boxed{\dfrac32 k^3 - \dfrac32 k^2 - k}[/tex]
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
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]__________________________
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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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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Please please help me. It’s due right now! it would mean a lot :))<3 Pls show work!!!
Answer: 9 triangles.
Step-by-step explanation:
Just make a dot in the middle and make lines and connect each line to an edge or bend.
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.)
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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Write y=5x-9 in standard form
Will give brainliest!
Answer:
5x - y = 9
Step-by-step explanation:
the equation of a line in standard form is
Ax + By = C ( A is a positive integer and B, C are integers )
given
y = 5x - 9 ( subtract 5x from both sides )
- 5x + y = - 9 ( multiply through by - 1 )
5x - y = 9 ← in standard form
A chemist wants to make 45 ml of a 17% acid solution by mixing a 13% acid solution and a 19 % acid solution. How many milliliters of each solution should the chemist use?
Answer:
22 millilitres of 13% solution
23 millilitres of 19% solution
Step-by-step explanation:
A + B = 45
13A + 19B = 45 x 17 = 765
-13A - 13B + 13A + 19B = -650 + 765 = 115
5B = 115
B = 23
22 + 23 = 45
A = 22
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
Consider the following pair of equations:
y = x + 4
y = −2x − 2
Explain how you will solve the pair of equations by substitution. Show all the steps and write the solution in (x, y) form.
Source
StylesFormatFontSize
Answer:
(-2, 2)
Step-by-step explanation:
Given system of equations:
[tex]\begin{cases}y=x+4\\y=-2x-2 \end{cases}[/tex]
To solve by substitution, substitute the first equation into the second equation:
[tex]\implies x+4=-2x-2[/tex]
Add 2x to both sides:
[tex]\implies x+4+2x=-2x-2+2x[/tex]
[tex]\implies 3x+4=-2[/tex]
Subtract 4 from both sides:
[tex]\implies 3x+4-4=-2-4[/tex]
[tex]\implies 3x=-6[/tex]
Divide both sides by 3:
[tex]\implies \dfrac{3x}{3}=\dfrac{-6}{3}[/tex]
[tex]\implies x=-2[/tex]
Substitute the found value of x into the first equation and solve for y:
[tex]\implies y=-2+4[/tex]
[tex]\implies y=2[/tex]
Therefore, the solution to the given system of equations is (-2, 2).
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Substitute y value from first eqn in second equation
y=-2x-2x+4=-2x-23x=-6x=-2Put in first one
y=-2+4=2(-2,2) is the solution
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
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.
For the geometric series 1 - 2/3 + 4/9 - 8/27....
find s8
Answer:
Step-by-step explanation:
The sum of an alternating geometric series SUM((-1)^n*ar^n) = a/(1+r). The given series has r=2/3 and a=1. The sum will be 1/(1+2/3)= 3/5
Hello,
We have s0 = 1 and q = -2/3
[tex]S _{n} = S _{0} \times q {}^{n} = 1 \times ( - \frac{2}{3} ) {}^{n} [/tex]
[tex]S _{8} = ( - \frac{2}{3} ) {}^{8} = \frac{2 {}^{8} }{3 {}^{8} } = \frac{256}{6 561} [/tex]
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.)