Given:
[tex]6\cos^2\theta+\cos\theta-2=0[/tex]Let
[tex]\cos\theta=t[/tex]Then
[tex]\begin{gathered} 6t^2+t-2=0 \\ (2t-1)(3t+2)=0 \\ 2t-1=0 \\ \Rightarrow t=\frac{1}{2} \\ \\ 3t+2=0 \\ \Rightarrow t=-\frac{2}{3} \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} \cos\theta=\frac{1}{2} \\ \\ \Rightarrow\theta=60^o \\ \\ \cos\theta=-\frac{2}{3} \\ \Rightarrow\theta=131.8^o \end{gathered}[/tex]Calculating number of periods?How long will an initial bank deposit of $10,000 grow to $23,750 at 5% annual compound interest?
For an initial amount P with an annually compounded interest rate r, after t years the total amount A is is given by:
[tex]A=P(1+r)^t[/tex]Then we have:
[tex]\begin{gathered} \frac{A}{P}=(1+r)^t \\ \ln\frac{A}{P}=t\ln(1+r) \\ t=\frac{\ln\frac{A}{P}}{ln(1+r)} \end{gathered}[/tex]For P = $10,000, A = $23,750 and r = 0.05, we have:
[tex]t=\frac{\ln\frac{23750}{10000}}{\ln(1+0.05)}\approx17.73\text{ years}[/tex]What's the sum of ten terms of a finite arithmetic series if the first term is 13 and the last term is 89?
The sum of the n first terms in an arithmetic series is given by the following formula
[tex]S_n=n\cdot(\frac{a_1+a_n}{2})[/tex]Where a_1 represents the first term, a_n represents the n-th term, and n the amount of terms we want to sum.
The first term of our sequence is 13, the tenth term is 89 and the amount of terms is 10. Plugging those values in our formula, we have
[tex]S_{10}=10\cdot(\frac{13+89}{2})=10\cdot51=510[/tex]This sum is equal to 510.
You want to enlarge a picture by a factor of 4.5 from its current size of 4 inches by 6 inches. What is the size of the enlarged picture?a. 18 in. by 27 in.b.8.5 in. by 10.5 in.c. 18 in. by 10.5 in.d. 8.5 in. by 27 in.
If we want to enlarge the picture by a factor of 4.5, the perimeter will also increase by the factor of 4.
[tex]\begin{gathered} \text{New dimension =}4.5\text{ (old dimension)} \\ \text{New dimension=4.5 (4 by 6)} \\ \text{New dimension=18 inches by 27 inches} \end{gathered}[/tex]Hence, the correct option is Option A
What is the equation of the line that passes through the given points (2,3) and (2,5)
Solution:
The equation of a line that passes through two points is expressed as
[tex]\begin{gathered} y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1) \\ where \\ (x_1,y_1)\text{ and} \\ (x_2,y_2)\text{ are the coordinates of the points } \\ through\text{ which the line passes} \end{gathered}[/tex]Given that the line passes through the points (2,3) and (2, 5), this implies that
[tex]\begin{gathered} x_1=2 \\ y_1=3 \\ x_2=2 \\ y_2=5 \end{gathered}[/tex]By substitution, we have
[tex]\begin{gathered} y-3=\frac{5-3}{2-2}(x-2) \\ \Rightarrow y-3=\frac{2}{0}(x-2) \\ multiply\text{ through by zero} \\ 0(y-3)=2(x-2) \\ \Rightarrow0=2x-4 \\ add\text{ 4 to both sides} \\ 0+4=2x-4+4 \\ \Rightarrow4=2x \\ divide\text{ both sides by the coefficient of x, which is 2} \\ \frac{4}{2}=\frac{2x}{2} \\ \Rightarrow x=2 \\ \end{gathered}[/tex]Hence, the equation of the line that passes through the given points (2,3) and (2,5) is
[tex]x=2[/tex]Directions: Solve the problems below on a separate sheet of paper. You will use a variety of strategies (drawingpictures, building multiple towers, area models, algorithms, and partial products method for division) to solvethe problems. Please submit your answer by writing a complete sentence that expresses the final answer.1. Books are on sale for $8. Peter has $25 in his wallet. How many books can he buy?
books are on sale for $8
Peter has $25 dollar in his wallet
let the numbers of book be x
so,
if a book cost $8
x number of books cost $25
lets put it into mathematical statement
1 = $8
x = $25
lets cross multiply
1 X 25 = 8 X x
25 = 8x
8x = 25
divide both sides by 8
8x/8 = 25/8
x = 25/8
x = 3.125
x = 3 (approximately)
recall, we say x is the numer of books
so,
the number of books peter can by with is $25 in his wallet is 3atement
1
simplified (-4+2i)(3-9i)
6 + 42i
Expanding the expression, by using FOIL acronym
(-4+2i)(3-9i)
-12+36i+6i-18i²
-12 +42i -18i² Remember i²= -1
-12 + 42i -18(-1)
-12 + 42i +18
6 + 42i
2) Now we have that complex number in the form a +bi
-
show that the triangles are similar by measuring the lengths of their sides and comparing the ratios of their corresponding sides
ANSWER
EXPLANATION
The ratio between corresponding sides of similar triangles is constant - in other words, the ratio between each pair of corresponding sides gives the same value.
As shown in the questions, the ratios between corresponding sides are,
[tex]\begin{gathered} \frac{DE}{AB}=\frac{3}{2}=1.5 \\ \frac{DF}{AC}=\frac{1.5}{1}=1.5 \\ \frac{EF}{BC}=\frac{2.4}{1.6}=1.5 \end{gathered}[/tex]Since the three ratios between corresponding sides are the same, 1.5, the triangles are similar.
x to the 9th power times x to the 5 power
What is 6 x 1/4 in the simplest form
Answer:
[tex]6\cdot\frac{1}{4}=\frac{3}{2}[/tex]Step-by-step explanation:
Divide 6 by 4:
[tex]6\cdot\frac{1}{4}=\frac{6}{4}=\frac{3}{2}[/tex]I need to find the composite function with these two equations. I also need to find the domain.
Recall that:
[tex](f\circ f)(x)=f(f(x)).[/tex]Therefore:
[tex](f\circ f)(x)=f(\sqrt[]{x+2})=\sqrt[]{\sqrt[]{x+2}+2}.[/tex]Now, the above function is well defined as long as x+2 remains positive, therefore, it is well defined as long as x is greater or equal to -2.
Answer: The domain is:
[tex]\lbrack-2,\infty).[/tex]The composition is:
[tex](f\circ f)(x)=f(\sqrt[]{x+2})=\sqrt[]{\sqrt[]{x+2}+2}.[/tex]A researcher wants to study the amount of protein in pet food. Which one of the following is most likely to give theresearcher more accurate results?-take a sample of cat foods alone-take a sample of dog foods alone-take a sample of all pet foods mixed together-divide the pet foods into two different groups, cat and dog, and take a sample from each group
He will need to take sample of at least two different sample of pet food in order to analyze it more accurate. So, the researcher should:
divide the pet foods into two different groups, cat and dog, and take a sample from each group.
Round $43,569.14 the nearest dollar
To find:
Round $43,569.14 the nearest dollar
Solution:
The number after the decimal is less than 50. So, the amount $43,569.14 rounded to the nearest dollar is $43,569.
Thus, the answer is $43569.
this temperature to Fahrenheil. 1.3 If 1 cm'- 1 ml and 1 000 cm -1 4. Determine the following: 1.3.1 How many cm' are in 875 ? 1.3.2 How many t are there in 35,853 cm'?
We will solve it as follows:
1.3.1: We transform liters to cubic centimeters:
[tex]x=\frac{875\cdot1000}{1}\Rightarrow x=875000[/tex]So, there are 875 000 cubic centimeters.
1.3.2: We transfrom cubic centimenters into liters:
[tex]x=\frac{1\cdot35853}{1000}\Rightarrow x=35.853[/tex]So, there are 35.853 liters.
A fence is purchased and constructed as shown. There are 250 feet of fence used for the chorale. Determine the values for x and y that will maximize the area. Round your answers to the nearest tenth if needed. Type the value for the x dimension in the first blank (you do not need to type x = , but label your answer). Type the value for y in the second blank (you do not need to type y =, but label your answer).
2x + 3y = 250
y = (250 - 2x)/3 (1)
S = x * y
= (-2/3x + 250/3)*x
= -2/3(x - 125/2)^2 + 125^2/6
x = 125/2
Replacing the value of x in (1)
y = 125/3
ZABC is a right angle.А2197032°Bс
Given that angle ABC is a right angle, then:
21° + x° + 32° = 90°
x = 90° - 21° - 32°
x = 37°
This corresponds to the option: subtract 21° and 32° from 90°, x = 37°.
Triangle MNO was reflected over the x-axis Given M(-5,-1)Find the coordinate M
When we perform the reflection of a figure over the x-axis, we just have to change the sign of the y-coordinate of each point, like this: P(x,y) -> P'(x,-y).
Then after a reflection of the triangle, the point M goes from (-5,-1) to (-5, 1)
Then the correct answer is the last option (-5, 1)
NO LINKS!! Use the method of substitution to solve the system. (If there's no solution, enter no solution). Part 11z
Answer:
smaller x value: -1,-8larger x value: 5,16The parenthesis part is already taken care of by the teacher.
=================================================
Explanation:
y is equal to x^2-9 and also 4x-4. We can equate those two right hand sides and get everything to one side like this
x^2-9 = 4x-4
x^2-9-4x+4 = 0
x^2-4x-5 = 0
Then we can use the quadratic formula to solve that equation for x.
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(-4)\pm\sqrt{(-4)^2-4(1)(-5)}}{2(1)}\\\\x = \frac{4\pm\sqrt{36}}{2}\\\\x = \frac{4\pm6}{2}\\\\x = \frac{4+6}{2} \ \text{ or } \ x = \frac{4-6}{2}\\\\x = \frac{10}{2} \ \text{ or } \ x = \frac{-2}{2}\\\\x = 5 \ \text{ or } \ x = -1\\\\[/tex]
Or alternatively
x^2-4x-5 = 0
(x-5)(x+1) = 0
x-5 = 0 or x+1 = 0
x = 5 or x = -1
------------------------------
After determining the x values, plug them into either original equation to find the paired y value.
Let's plug x = 5 into the first equation:
y = x^2-9
y = 5^2-9
y = 25-9
y = 16
Or you could pick the second equation:
y = 4x-4
y = 4(5)-4
y = 20-4
y = 16
We have x = 5 lead to y = 16
One solution is (x,y) = (5,16)
This is one point where the two curves y = x^2-9 and y = 4x-4 intersect.
If you repeat the same steps with x = -1, then you should find that y = -8 for either equation.
The other solution is (x,y) = (-1,-8)
Answer:
[tex](x,y)=\left(\; \boxed{-1,-8} \; \right)\quad \textsf{(smaller $x$-value)}[/tex]
[tex](x,y)=\left(\; \boxed{5,16} \; \right)\quad \textsf{(larger $x$-value)}[/tex]
Step-by-step explanation:
Given system of equations:
[tex]\begin{cases}y=x^2-9\\y=4x-4\end{cases}[/tex]
To solve by the method of substitution, substitute the first equation into the second equation and rearrange so that the equation equals zero:
[tex]\begin{aligned}x^2-9&=4x-4\\x^2-4x-9&=-4\\x^2-4x-5&=0\end{aligned}[/tex]
Factor the quadratic:
[tex]\begin{aligned}x^2-4x-5&=0\\x^2-5x+x-5&=0\\x(x-5)+1(x-5)&=0\\(x+1)(x-5)&=0\end{aligned}[/tex]
Apply the zero-product property and solve for x:
[tex]\implies x+1=0 \implies x=-1[/tex]
[tex]\implies x-5=0 \implies x=5[/tex]
Substitute the found values of x into the second equation and solve for y:
[tex]\begin{aligned}x=-1 \implies y&=4(-1)-4\\y&=-4-4\\y&=-8\end{aligned}[/tex]
[tex]\begin{aligned}x=5 \implies y&=4(5)-4\\y&=20-4\\y&=16\end{aligned}[/tex]
Therefore, the solutions are:
[tex](x,y)=\left(\; \boxed{-1,-8} \; \right)\quad \textsf{(smaller $x$-value)}[/tex]
[tex](x,y)=\left(\; \boxed{5,16} \; \right)\quad \textsf{(larger $x$-value)}[/tex]
In a direct variation, y = 18 when x = 6. Write a direct variation equation that shows therelationship between x and yWrite your answer as an equation with y first, followed by an equals signSubmit
When 27 is subtracted from the square of anumber, the result is 6 times the number. Findthe negative solution.
Given: A statement, "When 27 is subtracted from the square of a
number, the result is 6 times the number."
Required: To determine the number.
Explanation: Let the number be x. Then according to the question-
[tex]x^2-27=6x[/tex]Rearranging the equation as -
[tex]x^2-6x-27=0[/tex]The quadratic equation can be simplified as follows-
[tex]\begin{gathered} x^2-9x+3x-27=0 \\ x(x-9)+3(x-9)=0 \\ (x+3)(x-9)=0 \\ x=-3\text{ or }x=9 \end{gathered}[/tex]Final Answer: The negative solution is-
[tex]x=-3[/tex]which of the following is equivalent to the expression i^41?
The Solution:
Given:
[tex]i^{41}[/tex]Required:
Find the equivalent of the given expression.
[tex]i^{41}=i^{40}\times i^1=i[/tex]Answer:
[option A]
Solve the system of equations 2x - 3y = 4 and 9x - 8y = - 26 by combining the
equations.
[tex]\sf \Large \boxed{\sf +}\\ \sf \Large \boxed{\sf +}\\\\ \sf \Large \boxed{\sf 11x+-11y=-22}\\\\ 2x+9x-3y-8y=4-26\\Combine\\11x-11y=-22\\Simplify\\x-y=-2\\x=y-2\\Plug\ the\ value\ in\ the\ equation\\2(y-2)-3y=4\\2y-4-3y=4\\-y-4=4\\-y=8\\y=-8\\Solve\ for\ x\\9x-8(-8)=-26\\9x+64=-26\\9x=-90\\x=-10[/tex]
Find from first principles the derivative of f:x maps to (x+2)all squared
Given:
[tex]f(x)=(x+2)^2[/tex]Required:
To find the first principles
Explanation:
First principle,
[tex]\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex][tex]=\lim_{h\to0}\frac{(x+h+2)^2-(x+2)^2}{h}[/tex][tex]=\lim_{h\to0}\frac{x^2+(h+2)^2+2x(h+2)-x^2-4-4x}{h}[/tex][tex]=\lim_{h\to0}\frac{h^2+4+4h+2xh+4x-4-4x}{h}[/tex][tex]\begin{gathered} =\lim_{h\to0}\frac{h^2+4h+2xh}{h} \\ \\ =\lim_{h\to0}\frac{h(h+4+2x)}{h} \\ \\ =\lim_{h\to0}(h+4+2x) \\ =2x+4 \end{gathered}[/tex]Final Answer:
[tex]2x+4[/tex]Val measures the diameter of a ball as 14 inches. How many cubic inches of air does this ball hold, to thenearest tenth? Use 3.14 forn.The ball holds aboutcubic inches of air.
we know that
The volume of the sphere is equal to
[tex]V=\frac{4}{3}\cdot\pi\cdot r^3[/tex]In this problem we have
r=14/2=7 in ----> the radius is half the diameter
pi=3.14
substitute the given values
[tex]\begin{gathered} V=\frac{4}{3}_{}\cdot(3.14)\cdot(7^3) \\ V=1,436.0\text{ in\textasciicircum{}3} \end{gathered}[/tex]answer is 1,436.0 cubic inchesI need to solve this problem and name the concepts used in the problem
In a pie chart, the sum of the angles for each variable or item is 360 degrees. also, the total percentage is 100
Looking at each flavor,
27% chose Glazier freeze, = 27/100 * 360 = 97.2 degrees
25% chose Fierce grape = 25/100 * 360 = 90 degrees
15.5% chose Extreme Citrico = 15.5/100 * 360 = 55.8 degrees
13.5% chose Cool Blue = 13.5/100 * 360 = 48.6 degrees
11.5% chose Lemon ice = 11.5/100 * 360 = 41.4
We want to determine the degrees for others
Therefore,
97.2 + 90 + 55.8 + 48.6 + 41.4 + others = 360
333 + others = 360
others = 360 - 333
others = 27 degrees
The correct option is C
The concept used is converting the given percentages to degrees and equation them to 360 degrees
100% is equivalent to 360 degrees
A principal of S2400 is invested at 8.75% interest compounded annually How much will the investment be worth after 7 years?
Explanation
The question wants us to determine the amount $2400 will yield after 7 years if compounded annually at a rate of 8.75%
To do so, we will use the formula:
[tex]\begin{gathered} A=P(1+r)^t \\ where \\ P=\text{ \$2400} \\ r=8.75\text{ \%=}\frac{8.75}{100}=0.0875 \\ t=7 \end{gathered}[/tex]Thus, if we substitute the values above we will have
[tex]\begin{gathered} A=\text{ \$}2400(1+0.0875)^7 \\ A=\text{ }\$2400\lparen1.0875\rparen^7 \\ A=\text{ \$2400}\times1.79889 \\ A=\text{ \$4317.34} \end{gathered}[/tex]Therefore, after 7 years, the investment will be worth $4317.34
The size of a population of bacteria is modeledby the function P, where P(t) gives thenumber of bacteria and t gives the number ofhours after midnight for 0 < t < 10. Thegraph of the function P and the line tangent toP at t= 8 are shown above. Which of thefollowing gives the best estimate for theinstantaneous rate of change of P at t = 8?
Answer: The graph of the P(t) has been provided, we have to find the instantaneous slope of P(t) at t = 8:
[tex]Slope=m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}[/tex]Therefore we need two y values and two x values, which can be obtained as follows:
[tex]\begin{gathered} t=8 \\ \\ \therefore\Rightarrow \\ \\ x_1=t_1=8-0.1=7.9 \\ \\ y_1=P(t_1)=P(7.9) \\ \\ x_2=t_2=8+0.1=8.1 \\ \\ y_2=P(t_2)=P(8.1) \\ \\ \therefore\rightarrow \\ \\ Slope=\frac{P(8.1)-P(7.9)}{t_2-t_1}\rightarrow(1) \\ \end{gathered}[/tex]Equation (1) corresponds to the third, option, therefore that is the answer.
Use a calculator to find θ to the nearest tenth of a degree, if 0° < θ < 360° and sin θ = -0.9945
Solution:
Given:
[tex]\sin \theta=-0.9945[/tex]Using the inverse trigonometric function,
[tex]\begin{gathered} \theta=\sin ^{-1}(-0.9945) \\ \theta=-83.988 \\ \theta\approx-84.0^0\text{ to the nearest tenth} \end{gathered}[/tex]However, since the sine of the angle is negative, it shows that the angle is in the third or fourth quadrant.
Hence, the possible values of the angle are,
[tex]\begin{gathered} \theta=-84+360=276.0^0 \\ \theta=180-(-84)=264.0^0 \end{gathered}[/tex]Therefore, the value of the angle to the nearest tenth of a degree is 264.0 degrees or 276.0 degrees.
(3,-4); m=6 write an equation in slope intercept form for the line through the given point with the given slope
y= 6x-22
Explanation
Step 1
Let
slope=6
Point (3,-4)
to find the equation in slope intercept form use
[tex]\begin{gathered} y-y_0=m(x-x_0) \\ \text{where} \\ (x_0,y_0)\text{ are the coordinates of the known point} \end{gathered}[/tex]Step 2
Replace,
[tex]\begin{gathered} \text{the know point = (3,-4) so} \\ y-y_0=m\left(x-x_0\right) \\ y-(-4)=6(x-3) \\ y+4=6x-18 \\ \text{substract 4 in both sides} \\ y+4-4=6x-18-4 \\ y=6x-22 \end{gathered}[/tex]I hope this helps you
Please see the picture below. Indeed help with parts of the question
Given
[tex]\frac{(x-4)^2}{4}-\frac{y^2}{9}=1[/tex]Find
Values of a and b for this conic section
Explanation
As we know the standard equation for conic section is given by
[tex]\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1[/tex]where (h , k) be the vertex
vertices (h+a , k) and (h-a , k)
given equation can be rewrite as
[tex]\frac{(x-4)^2}{2^2}-\frac{y^2}{3^2}=1[/tex]on comparing , we get
a = 2 and b = 3
Final Answer
Therefore , the value of a = 2 and b = 3
The length of your step is 34 inches (in.). If you walk 10,000 steps in a day, how many feet (ft.) will you walk? ?
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
step length = 34 inches
walking = 10000 steps
Step 02:
feet to inches
1 feet = 12 inches
1 step --------------- 34 inches
10000 steps ------- x
1 * x = 10000 * 34
x = 340000
340000 inches * ( 1 feet / 12 inches)
28333.33 feet
The answer is:
You will walk 28333.33 feet .