Given;
Number of boys = 160
Number of girls = 240
Total number of students = 160 + 240 = 400
To find the percentage of all students that are girls, use the formula below:
[tex]\begin{gathered} \text{ \% of girls = }\frac{Number\text{ of girls}}{Total\text{ number}}\times\frac{100}{1} \\ \\ \text{ } \end{gathered}[/tex]Therefore, we have:
[tex]\begin{gathered} \text{ \% of girls = }\frac{240}{400}\times\frac{100}{1} \\ \\ \text{ = }0.6\text{ }\times\text{ 100 = 60\%} \end{gathered}[/tex]The percent of all the students that are girls is 60 percent.
ANSWER:
60%
Find anexpression which represents the sum of (-6x + 6) and (-3x – 7) insimplest terms.
We are asked to find the sum of the following expressions
[tex](-6x+6)\: \: and\: \: (-3x-7)[/tex]First of all, expand the parenthesis
[tex]\begin{gathered} (-6x+6)+\: (-3x-7) \\ -6x+6-3x-7 \end{gathered}[/tex]Now, collect the like terms together and add/subtract
[tex]\begin{gathered} -6x+6-3x-7 \\ (-6x-3x)+(6-7) \\ (-9x)+(-1)_{} \\ -9x-1 \end{gathered}[/tex]Therefore, the sum of the given expressions in the simplest form is
[tex]-9x-1[/tex]How to find the value of X in problem 15
We are asked to determine the value of "x" and "y".
To determine the value of "y" we will use the facto that since WP is a median this means that:
[tex]AP=PH[/tex]Substituting the values in terms of "y" we get:
[tex]3y+11=7y-5[/tex]Now, we solve for "y". To do that we will first subtract "7y" from both sides:
[tex]\begin{gathered} 3y-7y+11=7y-7y-5 \\ -4y+11=-5 \end{gathered}[/tex]Now, we subtract 11 from both sides:
[tex]\begin{gathered} -4y+11-11=-5-11 \\ -4y=-16 \end{gathered}[/tex]Now, we divide both sides by -4:
[tex]\begin{gathered} y=-\frac{16}{-4} \\ \\ y=4 \end{gathered}[/tex]therefore, the value of "y" is 4.
Now, to determine the value of "x" we will use the fact that since WP is an angle bisector we have that:
[tex]m\angle HWP+m\angle PWA=m\angle HWA[/tex]We also have the:
[tex]m\angle PWA=m\angle HWP[/tex]Therefore, we have:
[tex]\begin{gathered} m\operatorname{\angle}HWP+m\operatorname{\angle}HWP=m\operatorname{\angle}HWA \\ 2m\operatorname{\angle}HWP=m\operatorname{\angle}HWA \end{gathered}[/tex]Now, we substitute the values:
[tex]2(x+12)=4x-16[/tex]Now, we divide both sides by 2:
[tex]x+12=2x-8[/tex]Now, we subtract 2x from both sides:
[tex]\begin{gathered} x-2x+12=2x-2x-8 \\ -x+12=-8 \end{gathered}[/tex]Now, we subtract 12 from both sides:
[tex]\begin{gathered} -x+12-12=-8-12 \\ -x=-20 \\ x=20 \end{gathered}[/tex]This means that the value of "x" is 20.
To determine if WP is an altitude we need to determine if the angle APW is 90 degrees. To do that we use the fact that the sum of the interior angles of a triangle always adds up to 180, therefore:
[tex]m\angle WPA+m\angle PWA+m\angle PAW=180[/tex]We substitute the values in terms of "x":
[tex]m\angle WPA+(x+12)+(3x-2)=180[/tex]Now, we substitute the value of "x":
[tex]m\angle WPA+(20+12)+(3(20)-2)=180[/tex]Solving the operations:
[tex]m\angle WPA+90=180[/tex]now, we subtract 90 from both sides:
[tex]\begin{gathered} m\angle WPA=180-90 \\ m\angle WPA=90 \end{gathered}[/tex]Since WPA is 90 degrees and WP is a median and bisector this means that WP is an altitude.
The question is on the image below
Maximum number of identical boxes with no. of supply items in each box will be:
a. 78 boxes with 1 pencil and 1 eraser in each box.
b. 195 boxes with 1 notebook and 1 folder in each box.
c. 65 boxes with 1 eraser, 1 marker and 2 folders in each box.
First Lana will make 78 boxes with 1 pencil and 1 eraser in each box. After that she'll be left with
143 - 78 = 65 erasers.
Secondly she will make 195 boxes with 1 notebook and 1 folder in each box. After that she'll be left with
330 - 195 = 135 folders.
Next she will make 65 boxes with 1 eraser, 1 maker and 2 folders in each box.
By doing this she will be able to make maximum number of identical boxes.
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How many cubic feet of warehouse space are needed for 430 boxes 12in by 8in by 9in?
SOLUTION
From the question we want to know how many cubic-feet of a warehouse can contain 430 boxes, whereby "each one" of these 430 boxes measures 12 inches by 8 inches by 9 inches
Firstly we have to change these inches of the sides of ecah of these boxes to feet.
12 inches make a foot.
Hence each box in feet will measure
[tex]\begin{gathered} \frac{12}{12}ft\times\frac{8}{12}ft\times\frac{9}{12}ft \\ =1\times\frac{2}{3}\times\frac{3}{4} \\ =\frac{2}{4} \\ =\frac{1}{2}ft^3 \end{gathered}[/tex]So each boxes in feet will measure half cubic foot.
The warehouse that will contain 430 of these boxes should measure
[tex]\begin{gathered} 430\times\frac{1}{2} \\ =215ft^3 \end{gathered}[/tex]Hence, the answer is 215 cubic-feet
Graph AABC with A(4, 7), B(0,0), and C(8, 1).a. Which sides of AABC are congruent? How do you know?b. Construct the bisector of ZB. Mark the intersection of the ray and AC as D.c. What do you notice about AD and CD?
a) Two sides of a triangle are concruent when they are the same length. First calculate the lenght of each side
[tex]\begin{gathered} AC^2=\text{ (X\_c-X\_a)}^2+(Y_a-Y_c)^2=(8-4)^2+(7-1)^2=\text{ 52} \\ AC=\sqrt{52}=7.2 \end{gathered}[/tex][tex]\begin{gathered} AB^2=(X_a-X_b)^2+(Y_a-Y_b)^2=(4-0)^2+(7-0)^2=\text{ 65} \\ AB=\sqrt{65}=8.06\approx8 \end{gathered}[/tex][tex]\begin{gathered} BC^2=(X_c-X_b)^2+(Y_c-Y_b)^2=(8-0)^2+(1-0)^2=\text{ 65 } \\ BC=\sqrt{65}=8.06\approx8 \end{gathered}[/tex]Sides AB and BC aren congruent.
b)
The bisector divides the triangle in exact halves.
The bisector is the blue line, in green you'll se the length of each side.
c)
Hello, I need helping solving for x by completing the square.
EXPLANATION:
Given;
We are given the quadratic equation as shown below;
[tex]x^2-8x+13=0[/tex]Required;
We are required to solve for x by completing the square method.
Step-by-step solution;
We start with the constant 13.
Subtract 13 from both sides of the equation;
[tex]x^2-8x+13-13=0-13[/tex][tex]x^2-8x=-13[/tex]Next we take the coefficient of x (that is -8). We half it, and then square the result. After that we add it to both sides of the equation;
[tex]\begin{gathered} \frac{1}{2}\times-8=-\frac{8}{2} \\ Next: \\ (-\frac{8}{2})^2 \end{gathered}[/tex]We now have;
[tex]x^2-8x+(-\frac{8}{2})^2=-13+(-\frac{8}{2})^2[/tex]We can now simplify this;
[tex]x^2-8x+(-4)^2=-13+(-4)^2[/tex][tex]x^2-8x+16=-13+16[/tex][tex]x^2-8x+16=3[/tex]We now factorize the left side of the equation;
[tex]\begin{gathered} x^2-4x-4x+16 \\ (x^2-4x)-(4x-16) \\ x(x-4)-4(x-4) \\ (x-4)(x-4) \\ Therefore: \\ (x-4)^2 \end{gathered}[/tex]we can now refine our equation to become;
[tex](x-4)^2=3[/tex]We can now solve for x as follows;
Take the square root of both sides;
[tex]x-4=\pm\sqrt{3}[/tex]Therefore;
[tex]\begin{gathered} x-4=\sqrt{3} \\ x=\sqrt{3}+4 \\ Also: \\ x-4=-\sqrt{3} \\ x=-\sqrt{3}+4 \end{gathered}[/tex]ANSWER:
[tex]\begin{gathered} x_1=4+\sqrt{3} \\ x_2=4-\sqrt{3} \end{gathered}[/tex]There are 8 triangles and 20 circles. What is the simplest ratio of triangles to circles?
Answer:
2:5
Step-by-step explanation:
8:20
= 4:10 (simplifying)
= 2:5
Answer:
2:5
Step-by-step explanation:
8=2*2*2, 20=2*2*5
cancel out the numbers they have in common
8=2*2*2, 20=2*2*5
=2,5
as a ratio
2:5
Erica is given the diagram below and asked to prove that AB DF. What would be the missing step of the proof? Given: Point B is the midpoint of EF, and point A is the midpoint of ED. Prove: AB DF
Given
To find the missi
F (x)=x^2+4 what is f(-4)
ANSWER
f(-4) = 20
EXPLANATION
To find f(-4) we just have to replace x by -4 in function f(x):
[tex]f(-4)=(-4)^2+4[/tex]First solve the exponents. Remember that if the exponent is even and the result is always positive, either the base is positive or negative:
[tex]f(-4)=16+4=20[/tex]36\100 as a percentage
Notice that in the fraction
[tex]\frac{36}{100}[/tex]Can be interpreted as "36 out of every 100"
As a percentage, this means 36%
Simplify. Final answer should be in standard form NUMBER 18
4(2 - 3w)(w^2 - 2w + 10) =
(8 - 12w)(w^2 - 2w + 10) =
8w^2 - 16w + 80 - 12w^3 + 24w^2 - 120w =
- 12w^3 + 32w^2 - 123w + 80
9. Find the volume of the triangular pyramid. (2pts)-10 mI9 m16 m
Answer:
240 m³
Explanation:
The volume of a pyramid is equal to:
[tex]V=\frac{1}{3}\times B\times H[/tex]Where B is the area of the base and H is the height of the pyramid.
Then, the base of the pyramid is a triangle, so the area of a triangle is equal to:
[tex]B=\frac{b\times h}{2}[/tex]Where b is the base of the triangle and h is the height of the triangle. So, replacing b by 16 m and h by 9 m, we get:
[tex]B=\frac{16\times9}{2}=\frac{144}{2}=72m^2[/tex]Finally, replacing B by 72 m² and H by 10 m, we get that the volume of the pyramid is equal to:
[tex]V=\frac{1}{3}\times72\times10=\frac{1}{3}\times720=240m^3[/tex]Therefore, the volume is 240 m³
Complete the coordinate proof. Answer choices are on the bottom.
Given:
There are given that the triangle, ABC.
Where:
[tex]\begin{gathered} A=(3,6) \\ B=(5,0) \\ C=(1,0) \end{gathered}[/tex]Explanation:
According to the question, we need to prove that the isosceles triangle:
So,
From the concept of the isosceles triangle:
The isosceles triangle is defined when two sides of the length of any triangle are equal.
Then,
First, we need to find the length of the sides by using the distance formula:
So,
[tex]\begin{gathered} AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ AB=\sqrt{(5-3)^2+(0-6)^2} \\ AB=\sqrt{(2)^2+(-6)^2} \\ AB=\sqrt{4+36} \\ AB=\sqrt{40} \end{gathered}[/tex]Then,
[tex]\begin{gathered} AC=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ AC=\sqrt{(1-3)^2+(0-6)^2} \\ AC=\sqrt{(-2)^2+(-6)^2} \\ AC=\sqrt{4+36} \\ AC=\sqrt{40} \end{gathered}[/tex]And,
[tex]\begin{gathered} CB=\sqrt{(5-1)^2+(0-0)^2} \\ CB=\sqrt{(4)^2+0} \\ CB=4 \end{gathered}[/tex]Final answer:
Hence, the step and values of the sides are shown below;
[tex]\begin{gathered} CA=d \\ AB=d \end{gathered}[/tex]And,
The side CA and AB is congruence by the definition of h
And,
Triangle ABC is an isosceles triangle by the the defination of b.
Use the figure to find the measures of the numbered angles. 95 23 24 = Explain your reasoning.
The given angle and angle 3 are corresponding angles, that is, angles that are on the same corner at each intersection. Graphically,
Corresponding angles are congruent, so
[tex]\angle3=95\text{\degree}[/tex]On the other hand, angle 3 and angle 4 are supplementary angles, that is, add up to 180°. Graphically,
[tex]A+B=180\text{\degree}[/tex]So, you have
[tex]\begin{gathered} \angle3+\angle4=180\text{\degree} \\ 95\text{\degree}+\angle4=180\text{\degree} \\ \text{ Subtract 95\degree from both sides of the equation} \\ 95\text{\degree}+\angle4-95\text{\degree}=180\text{\degree}-95\text{\degree} \\ \angle4=85\text{\degree} \end{gathered}[/tex]Therefore, the measures of the numbered angles are
[tex]\begin{gathered} \angle3=95\text{\degree} \\ \angle4=85\text{\degree} \end{gathered}[/tex]2-18 72 20=34-To=315)-10=35 5)EXTENSION: a) In right A DEF, m D = 90 and mZF is 12 degrees less than twice mze. Find mZE. b) in AABC, the measure of ZB is 21 less than four times the measure of LA, and the measure of ZC is 1 more than five times the measure of ZA. Find the measure, in degrees, of each angle of ABC.
As given by the question
There are given that in the right triangle DEF, angle D is 90 degrees and angle f is 12 degrees less than angle E.
Now,
The sum of the three measures of a triangle is always 180 degree
So,
[tex]m\angle D+m\angle E+m\angle F=180[/tex]Where angle D is 90 degree
Then,
[tex]\begin{gathered} m\angle D+m\angle E+m\angle F=180 \\ 90+m\angle E+m\angle F=180 \\ m\angle E+m\angle F=90 \end{gathered}[/tex]Also we are given that
[tex]\begin{gathered} F+12=2E \\ F=2E-12 \end{gathered}[/tex]Therefore, substituting for F back into E+F=90
Then,
[tex]\begin{gathered} E+(2E-12)=90 \\ 3E-12=90 \\ 3E=102 \\ E=34 \end{gathered}[/tex]So, angle E is 34 degrees, which is the answer.
ABCD is a parallelogram Find m angle C.В,11492x + 1234A4
Given the parallelogram ABCD:
The sum of every two adjacent angles = 180
so,
m∠B + m∠C = 180
m∠C = 180 - m∠B = 180 - 114 = 66
So, the answer will be m∠C = 66
Convert the radical to exponential form. Assume variables represent positive real numbers.
Exponential Form of Radicals
A radical can be expressed in exponential form by using the equivalence:
[tex]\sqrt[m]{x^n}=x^{\frac{n}{m}}[/tex]We are given the expression:
[tex]\sqrt[4]{16a^4b^3}[/tex]It can be separated into several radicals:
[tex]\sqrt[4]{16a^4b^3}=\sqrt[4]{16}\cdot\sqrt[4]{a^4}\cdot\sqrt[4]{b^3}[/tex]Now we apply the equivalence on each individual radical:
[tex]\begin{gathered} \sqrt[4]{16a^4b^3}=\sqrt[4]{2^4}\cdot\sqrt[4]{a^4}\cdot\sqrt[4]{b^3} \\ \sqrt[4]{16a^4b^3}=2^{\frac{4}{4}}\cdot a^{\frac{4}{4}}\cdot b^{\frac{3}{4}} \end{gathered}[/tex]Simplifying:
[tex]\sqrt[4]{16a^4b^3}=2ab^{\frac{3}{4}}[/tex]1 8. Dee Saint earns a monthly salary of $750 plus a 6% commission on all sales over $1,000 each month. Last month, her sales were $5,726. What was her income for the month?
Her monthly income for last month was $1,093.56
Here, we want to calculate the monthly income for Dee Saint
Mathematically, from the information given in the question, we can have this as;
[tex]\begin{gathered} 750\text{ + 6\% of \$5726} \\ \\ =\text{ 750 + 0.06(5726)} \\ \\ =\text{ 750 + 343.56} \\ \\ =\text{ \$1,093.56} \end{gathered}[/tex]Let P(x)=6x and Q(x)=2x^3 + 3x^2 + 1. Find P(x)⋅Q(x)
Explanation
We are given the following functions:
[tex]\begin{gathered} P(x)=6x \\ Q(x)=2x^3+3x^2+1 \end{gathered}[/tex]We are required to determine the following:
[tex]P(x)\cdot Q(x)[/tex]This is achieved thus:
[tex]\begin{gathered} P(x)=6x \\ Q(x)=2x^3+3x^2+1 \\ \\ \therefore P(x)\cdot Q(x)=(6x)(2x^3+3x^2+1) \\ P(x)\cdot Q(x)=6x\cdot2x^3+6x\cdot3x^2+6x\cdot1 \\ P(x)\cdot Q(x)=12x^4+18x^3+6x \end{gathered}[/tex]Hence, the answer is:
[tex]\begin{equation*} 12x^4+18x^3+6x \end{equation*}[/tex]8Suppose Z follows the standard normal distribution. Use the calculator provided, or this table, to determine the value of c so that the following is true.p=(-c ≤ Z ≤ c ) =0.9127Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.
The value of c such that [tex]P(-c\leq Z\leq c)=0.9127[/tex] is true is 0.0873 where Z follows the standard normal distribution.
It is given to us that -
[tex]P(-c\leq Z\leq c)=0.9127[/tex] is true
It is also given that Z follows the standard normal distribution.
We have to find out the value of c.
Since Z follows the standard normal distribution, so we can say that
Z ∼ N(0,1)
To find out c,
[tex]P(-c\leq Z\leq c)=0.9127\\= > P(Z\leq c)-P(Z\leq -c)=0.9127\\[/tex]
Since there is a symmetric z-distribution, the above equation can be represented as -
[tex][1-P(Z\leq -c)]-P(Z\leq -c) = 0.9127\\= > 1-P(Z\leq -c) - P(Z\leq -c) = 0.9127\\= > 1-2P(Z\leq -c)=0.9127\\= > 2P(Z\leq -c)=0.0873\\= > P(Z\leq -c)=0.04365[/tex]
=> -c ≈ 0.0873 (Using online calculator)
Therefore, the value of c such that [tex]P(-c\leq Z\leq c)=0.9127[/tex] is true is 0.0873 where Z follows the standard normal distribution.
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Answer:
The value of c such that is true is 0.0873 where Z follows the standard normal distribution.
Step-by-step explanation:
For each ordered pair, determine whether it is a solution to the sytem of equations.
Given
We have the system of equations:
[tex]\begin{gathered} 3x\text{ - 2y = -4} \\ 2x\text{ + 5y = -9} \end{gathered}[/tex]The ordered pair that would be a solution to the given system of equations must satisfy both equations. There can only be one ordered pair and this can be obtained by solving the system of equations simultaneously
Using a graphing tool, the plot of the lines is shown below:
The point where the lines intercept is the solution to the system of equations.
Hence the ordered pair that is a solution is (-2, -1)
Answer:
(4,8) - No
(8, -5) - No
(0, 3) - No
(-2, -1) - Yes
I have taken a picture of the question. Thank you.
The original width of the rectangular piece of metal is 21 inches.
let us take into consideration the width of the rectangle be x,
Length is given to be 5 inches more than the width
∴ length = x+5
Now squares of side 1 inch is cut from all the corners of the rectangle to form a box in the form of a cuboid.
New length of the base of the box = (x+5) - 2 = x+3 inches
new width of the box = x -2 inches
Height of the box = 1 inch
Volume of the box that is formed
= (x+3 ) · (x -2) × 1
= x² - x - 6
The given volume of the box is 414 cubic inches
Therefore:
x² - x - 6 = 414
or, x² - x -420 = 0
Solving the quadratic equation by middle term factorization we get :
or, ( x - 21 ) ( x + 20 ) = 0
Now either x=-20( not possible)
or , x =21 inches.
Therefore the original width of the rectangle is 21 inches.
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A ladder 7.90 m long leans against the side of a building. If the ladder is inclined at an angle of 74.5° to the horizontal, what is the horizontal distance from the bottom of the ladder to the building?____________ m
First, let's sketch the problem:
To find the horizontal distance d, we can use the cosine relation of the angle 74.5°.
The cosine is equal to the length of the adjacent leg to the angle over the length of the hypotenuse.
So we have:
[tex]\begin{gathered} \cos74.5°=\frac{d}{7.9}\\ \\ 0.2672=\frac{d}{7.9}\\ \\ d=0.2672\cdot7.9\\ \\ d=2.11\text{ m} \end{gathered}[/tex]Ingrid deposits $10,000 in an IRA. What will be the value of her investment in 6 years if the investment is earning 3.2% per year and is compounded continuously? Round to the nearest cent.
We have a initial deposit of $10,000 (PV=10,000).
The investment last 6 years (t=6).
The annual interest rate is 3.2% (r=0.032) and is compounded continously.
The equation to calculate the future value FV of the inverstment for this conditions is:
[tex]\begin{gathered} FV=PV\cdot e^{rt} \\ FV=10,000\cdot e^{0.032\cdot6} \\ FV=10,000\cdot e^{0.192}. \\ FV\approx10,000\cdot1.2116705 \\ FV\approx12,116.71 \end{gathered}[/tex]The value of her investment will be $12,116.71.
Which is the greatest number?A. 50 – 16piB. 16 - sqrt(410)C. -sqrt(20) + 1/2D. 7/3 - (7pi/3)فر
First, we need to develop each case or take care of the following:
One number is greater than another if it is more at the right of the Real Line.
A negative number is lower than a positive number.
Between two negative numbers, the greater is the one near to zero.
Let develop the numbers:
A. 50 - 16pi is approximately -0.265472
B. 16 - sqrt(410) approximately equals to -4.24845
C. -sqrt(20) + 1/2 is approximately equals to -3.97213
D. 7/3 - (7*pi)/3 is approximately equaled to -4.99705
So taking into account the previous reasons at the beginning, we have that the number near to zero is -0.265472, which is the first option. Option A.
what is the GCF of 20 and 32
Given the following numbers
[tex]20,32[/tex]To find the greatest common factor, G.C.F.
The factor that can divide through two or more numbers evenly is the G.C.F
The factors of 20 and 32 are as follows
[tex]\begin{gathered} 20\Rightarrow1\times2\times2\times5 \\ 32\Rightarrow1\times2\times2\times2\times2\times2 \end{gathered}[/tex]The common factors between 20 and 32 is
[tex]\begin{gathered} \text{Common factors }=2,2 \\ G\mathrm{}C\mathrm{}F=2\times2=4 \\ G\mathrm{}C\mathrm{}F=4 \end{gathered}[/tex]Hence, the GCF of 20 and 32 is 4
Alternatively
Finding the G.C.F using table to find the G.C.F of 20 and 32
Therefore, the G.C.F is
[tex]G.C.F\Rightarrow2\times2=4[/tex]Hence, the G.C.F of 20 and 32 is 4
Mr. Fowler's science class grew two different varieties of plants as part of anexperiment. When the plant samples were fully grown, the studentscompared their heights.PlantvarietyHeight of plant(inches)20, 17, 19, 18, 21Mean Mean absolute deviation(Inches)Variety A191.2Variety B13, 18, 11,9,14132.4Based on these data, which statement is true?O A. The maximum height for plants from variety B is greater than forvariety A.B. Plants from variety A always grow taller than plants from variety B.C. The height of a plant from variety B is likely to be closer to themean.D. The height of a plant from variety A is likely to be closer to themean.
Let's analyze all the statements and see why they are false or true.
A. FALSE
The tallest plant in variety B is just 18 tall, while the variety A we have 21.
B. FALSE
We do have plants in A that have the same height as B.
C. FALSE
The standard deviation measure how far it's from the mean, the variety B has a 2.4 standard deviation, which means that the height can be more distant from the mean than in variety A.
D. True
Justified by C. Variety A has a 1.2 standard deviation, which means it's more likely to be closer to the mean
What is the solution of the inequality shown below? 1 + a 4 enter the correct answer
We are given the following inequality:
[tex]1+a\le4[/tex]To solve this inequality we will subtract 1 to both sides:
[tex]\begin{gathered} 1-1+a\le4-1 \\ a\le3 \end{gathered}[/tex]And thus we get the solution.
The number of visits to public libraries increased from 1.2 billion in 1990 to 1.6 billion in 1994. Find the average rate of change in the number of public library visits from 1990 to 1994.
Okay, here we have this:
Considering the provided information, we are going to calculate the requested rate of change, so we obtain the following:
We will replace in the rate of change formula with the following points: (1990, 1.2) and (1994, 1.6), then we have:
Rate of change=(f(b)-f(a))/(b-a)
Rate of change=(1.6-1.2)/(1994-1990)
Rate of change=0.4/4
Rate of change=0.1 Billion
Finally we obtain that the average rate of change in the number of public library visits from 1990 to 1994 is 0.1 billion.
Two weather stations are aware of a thunderstorm located at point C. The weather stations A and B are 24 miles apart.
Assuming the dashed lines are parallel and perpendicular to the base, we can start by draw a third parallel line that passes through C and naming some distances:
Now, we can see that the given angles are alternate interior angles with respect to the angles formed by the new perpendicular line and the lines AC and BC:
Now, we can see that b and the base a + 24 are related with the tangent of 48°:
[tex]\tan 48\degree=\frac{a+24}{b}[/tex]Also, b and a are related with the tangent of 17°:
[tex]\tan 17\degree=\frac{a}{b}[/tex]We can solve both for b and equalize them:
[tex]\begin{gathered} b=\frac{a+24}{\tan48\degree} \\ b=\frac{a}{\tan17\degree} \\ \frac{a+24}{\tan48°}=\frac{a}{\tan17\degree} \\ a\tan 17\degree+24\tan 17\degree=a\tan 48\degree \\ a\tan 48\degree-a\tan 17\degree=24\tan 17\degree \\ a(\tan 48\degree-\tan 17\degree)=24\tan 17\degree \\ a=\frac{24\tan17\degree}{\tan48\degree-\tan17\degree}=\frac{24\cdot0.3057\ldots}{1.1106\ldots-0.3057\ldots}=\frac{7.3375\ldots}{0.8048\ldots}=9.1162\ldots \end{gathered}[/tex]Now, we can relate a and x with the sine of 17°:
[tex]\begin{gathered} \sin 17\degree=\frac{a}{x} \\ x=\frac{a}{\sin17\degree}=\frac{9.1162\ldots}{0.2923\ldots}=31.18\ldots\approx31.2 \end{gathered}[/tex]And x is the distance between A and C, the storm. Thus the answer is approximately 31.2 miles, fourth alternative.