Given:
The population P of a city is given by,
[tex]P=115600e^{0.024t,}[/tex]To find:
The time taken for the population to reach 130,000.
Explanation:
Substituting P = 130,000 in the given function, we get
[tex]\begin{gathered} 130000=115600 \\ e^{0.024t}=\frac{130000}{115600} \\ e^{0.024t}=1.1245 \\ 0.024t=\ln1.1245 \\ 0.024t=0.1174 \\ t=4.891 \\ t\approx4.89years \end{gathered}[/tex]Therefore, the number of years required for the population to reach 130,000 is 4.89years.
Final answer:
The number of years required is 4.89years.
Need to find equation f(x) = a(b)^xfor these 2 sets of points.(0, -3) (1, -³/2)
y = a b^x
Substitute the first set of points into the equation
-3 = a * b^0
-3 = a * (1)
-3 = a
y = (-3)* b^ x
Now using the second point
-3/2 = -3 * ( b)^1
-3/2 = -3 *b
Divide each side by -3
1/2 = b
y = -3 ( 1/2) ^x
During a 60 minute boxing class Abby burns 480 calories, on a 40 minute run, Abby burns 440 calories, while cycling for 30 minutes which exercise is the most efficient at burning calories?
Explanation:
Given that;
In a 60 minute boxing class Abby burns 480 calories.
The rate at which calories are burn during boxing is;
[tex]\begin{gathered} r_1=\frac{480}{60} \\ r_1=8\text{ calories/minute} \end{gathered}[/tex]Also;
on a 40 minute run,Abby burns 440 calories,
The rate at which calories are burn during cycling is;
[tex]\begin{gathered} r_2=\frac{440}{40} \\ r_2=11\text{calories/minute} \end{gathered}[/tex]while cycling for 30 minutes,
From the above rate the most efficient exercise at burning calories is the exercise with the highest rate of burning calories.
This is my math homework, I don’t understand how to find the seconds of the ball hit the ground
When the ball hits the ground, the height of the ball to the ground is h(t) = 0.
Therefore, we can now substitute h(t), and solve the function using quadratic formula.
[tex]\begin{gathered} h(t)=-16t^2+16t+400 \\ 0=-16t^2+16t+400 \\ \\ \text{The function is now in standard form where} \\ a=-16,b=16,c=400 \end{gathered}[/tex]Using the quadratic formula, substitute the following values a,b, and c.
[tex]\begin{gathered} t=\frac{ -b \pm\sqrt{b^2 - 4ac}}{ 2a } \\ t=\frac{ -16 \pm\sqrt{16^2 - 4(-16)(400)}}{ 2(-16) } \\ t=\frac{-16\pm\sqrt[]{256-(-25600)}}{-32} \\ t=\frac{ -16 \pm\sqrt{25856}}{ -32 } \\ t=\frac{ -16 \pm16\sqrt{101}\, }{ -32 } \\ \\ t=\frac{-16+16\sqrt{101}}{-32} \\ t=-4.52494 \\ \\ t=\frac{-16-16\sqrt[]{101}\, }{-32} \\ t=5.52494 \end{gathered}[/tex]We have two solutions, t = -4.52494, and t = 5.52494. However, we will disregard the negative time value.
Therefore, the ball will hit the ground after 5.52494 seconds.
Please see attachment for question. I have also uploaded a example for reference.
A=$88,000
P=$70,586
r=9.8% (0.098)
[tex]88,000=70,586e^{(0.098)t}[/tex]Divide both sides by 70,586:
[tex]\begin{gathered} \frac{88,000}{70,586}=\frac{70,586}{70,586}e^{(0.098)t} \\ \\ \frac{88,000}{70,586}=e^{(0.098)t} \end{gathered}[/tex]Find ln of both sides:
[tex]\begin{gathered} \ln (\frac{88,000}{70,586})=\ln (e^{(0.098)t}) \\ \\ \ln (\frac{88,000}{70,586})=0.098t \end{gathered}[/tex]Divide both sides by 0.098:
[tex]\begin{gathered} \frac{\ln (\frac{88,000}{70,586})}{0.098}=\frac{0.098}{0.098}t \\ \\ \frac{\ln(\frac{88,000}{70,586})}{0.098}=t \\ \\ \\ t=2.25 \end{gathered}[/tex]Then, t is 2.25 yearsFind x1) -4x=362) x+6=133) -9x=36
1) -4x=36
2) x+6=13
3) -9x=36
SolutionNumber 1[tex]\begin{gathered} -4x=36 \\ divide\text{ both sides by -4} \\ -\frac{4x}{-4}=\frac{36}{-4} \\ \\ x=-9 \end{gathered}[/tex]Number 2[tex]\begin{gathered} x+6=13 \\ collect\text{ the like terms} \\ x=13-6 \\ x=7 \end{gathered}[/tex]Number 3[tex]\begin{gathered} -9x=36 \\ divide\text{ both sides by -9} \\ -\frac{9x}{-9}=\frac{36}{-9} \\ \\ x=-4 \end{gathered}[/tex]two cyclists, 112 miles apart riding toward each other at the same time. One cycles 3 times as fast as the other. if they meet 4 hours later, what is the speed (in mi/h) of the faster cyclist?
distance= 112 miles apart
time = 4 hours
Speed of a = 3 *speed of b
Distance covered by a = 3 * distance covered by b
Speed = distance / time
3b+b = 112
4b = 112
b= 112/4
b= 28 miles
a = 3b
a = 3 *28
a= 84 miles
Since both travelled for 4 hours:
Speed of A = 84 /4 = 21 mph
Speed of b= 28 /4 = 7 mph
Speed of the faster cyclist : 21 mph
finding slop on the line
As per given diagram:
Take two points on the line (1,1) and (0,4)
For the slope of the line:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Now put the values in the formula:
[tex]\begin{gathered} m=\frac{4-1}{0-1} \\ m=\frac{3}{-1} \\ m=-3 \end{gathered}[/tex]So the slopw of the given line is -3.
find its volume. Use 3.14 as the approximate value of a25 m
we know that
the solid of the figure is a sphere
the volume of the sphere is equal to
[tex]V=\frac{4}{3}\cdot\pi\cdot r^3[/tex]we have
pi=3.14
r=25 m
substitute
[tex]\begin{gathered} V=\frac{4}{3}\cdot3.14\cdot25^3 \\ V=65,416.67\text{ m\textasciicircum{}3} \end{gathered}[/tex]the volume of the sphere is 65,416.67 cubic metersAn escalator at a shopping center is 200 ft long and has a vertical rise of 52 feet.What is the measure of the angle formed by the escalator and the ground? Round to the nearest degree
A right triangle is made, with measure:
From definition:
sin(α) = opposite/hypotenuse
From the picture: opposite to α is side of 52 long and the hypotenuse is 200 ft long. Then:
sin(α) = 52/200
sin(α) = 0.26
α = arcsin(0.26)
α = 15°
how far does a person travel in ft? this word problem is a little confusing but i understood everything else before this question.
Given:
The vertical height is 30ft.
The angle of elevation is 30 degrees.
To find:
The distance travelled by the person from bottom to top of the escalator.
Explanation:
Let x be the slant distance.
Since it is a right triangle.
Using the trigonometric ratio formula,
[tex]\begin{gathered} \sin\theta=\frac{Opp}{Hyp} \\ \sin30^{\circ}=\frac{30}{x} \\ \frac{1}{2}=\frac{30}{x} \\ x=30\times2 \\ x=60ft \end{gathered}[/tex]Therefore, the distance travelled by the person from the bottom to the top of the escalator is 60ft.
Final answer:
The distance travelled by the person from bottom to top of the escalator is 60ft.
What is the vertex of the absolute value function below?
Step-by-step explanation:
come on !
the vertex is typically a corner or a (general or local) extreme value or turning point of the curve.
so, what can that point be ?
there is only one candidate : (4, 1).
that means x = 4, y = 1.
that is the 4th answer option.
Triangle ABC , m∠A = 75°, m∠B = 65°, a = 23.5ftFind the length of the shortest side.Round to one decimal.
Given in triangle ABC , m∠A = 75°, m∠B = 65°, a = 23.5ft.
We have to find the third angle,
[tex]m\angle C=180-75-65=40[/tex]The shortest angle is angle C. So, the shortest side will be opposite to angle C.
Use the sine rule, to find the third side as follows:
[tex]\begin{gathered} \frac{\sin A}{a}=\frac{\sin C}{c} \\ \Rightarrow\frac{\sin75}{23.5}=\frac{\sin 40}{c} \\ \Rightarrow\frac{0.966}{23.5}=\frac{0.643}{c} \\ \Rightarrow0.0411=\frac{0.643}{c} \\ \Rightarrow c=\frac{0.643}{0.0411} \\ \Rightarrow c=15.6 \end{gathered}[/tex]Thus. the length of the shortest side is 15.6 ft.
A person can join The Fitness Center for $50. A member can rent the tennis ball machine for $10 an hour. Write a linear function to model the relationship between the number of hours the machine is rented (x) and the total cost (y).
(a) write a equation for this problem
(b) what is the initial value
(c) Determine the cost for renting the tennis ball machine for 2 hours? for 5 hours? and 0 hours?
(d) How many hours did a member rent the tennis ball machine if the total cost was $130?
a) An equation for the given problem is;
b) The initial value for the given problem is;
c) The number of hours that a member rented the tennis ball machine if the total cost was $130 is; 4 hours
How to solve a Linear equation model?The general formula for the equation of a line in slope intercept form is;
y = mx + c
where;
m is slope
c is y-intercept
a) Base price for joining the fitness center is $50 and as such this can be said to be the y-intercept. Then, the tennis ball machine is rented for $10 per hour.
Now, the number of hours the machine is rented (x) and the total cost (y), then the equation is;
y = 10x + 50
b) The initial value is the price when x = 0 which is the y-intercept and as such it is $50.
c) The cost for renting the tennis ball machine for 2 hours is;
y = 10(2) + 50
y = $70
The cost for renting the tennis ball machine for 5 hours is;
y = 10(5) + 50
y = $120
The cost for renting the tennis ball machine for 0 hours is;
y = 10(0) + 50
y = $50
d) If total cost is $130, then we have;
130 = 10x + 50
10x = 80
x = 80/10
x = 8 hours
Read more about Linear Equation model at; https://brainly.com/question/4074386
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Hello hope you are doing well. Can you help me with this please
To find Mario's current grade is to find the average of all his grades for the first quarter.
The average of his grade for the first quarter is the mean which is 70 percent.
Also, 70 percent is equivalent to a C minus
When information is presented in the form of a bar graph or time- series graph, you could get more exact values if all the data were just listed out in table form. Then why not always do that. Why bother with graphs?
Step 1:
Graphs are essentially a visual display of quantitative information along two axes. Visuals are used as a way for our brains to quickly understand information, which is a powerful tool if used correctly. Graphs can show a large amount of data quickly in a way that is easy to process, without distracting people with a bunch of numbers.
May I please get help with this I need help with finding the original Final point. I also need help with the graphing part, I am confused as to how I should do it after many tries Or where I should reflect it
We need to reflect the given figure across the y-axis. Then, we need to write the coordinates of the original and the reflected large dot.
When we reflect a point across the y-axis, its y-coordinate remains the same, and its x-coordinate changes the sign:
[tex](x,y)\rightarrow(-x,y)[/tex]The original coordinates of the large dot is: (5,3).
Thus, the coordinates after the reflection will be: (-5,3).
Doing the same with the coordinates of the other vertices and joining them, we obtain the reflected figure, as shown below:
Find the equation for a polynomial f(x) that satisfies the following:Degree 5- Root of multiplicity 1 at 2 = 1- Root of multiplicity 2 at x = 2- Root of multiplicity 2 at x = -2y-intercept of (0,–32)
The equation for this polynomial is:
[tex]\begin{gathered} 2(x-1)(x-2)^2(x+2)^2 \\ 2x^5-2x^4-16x^3+16x^2+32x-32 \end{gathered}[/tex]So that's the equation we're asking for.
Both could be the answers. However, this is the final one:
[tex]2x^5-2x^4-16x^3+16x^2+32x-32[/tex]Stephen began a baseball card collection by purchasing some cards. He increase the number of cards in the collection by a constant amount each week after that. The table below shows the total number of cards in the collection at the end of several weeks. Weeks completed since initial purchase :• 3• 6• 11Number of cards in the collection: •285•420•645How many cards did Steve initially purchase to the beginning of his collection?
The situation can be represented by a linear function, which is represented by the following expression:
[tex]\begin{gathered} y=mx+b \\ \text{Where, } \\ m=\text{slope} \\ b=y-\text{intercept} \end{gathered}[/tex]Since he increased the number of cards by a constant amount each week, that means we have proportionality:
[tex]\begin{gathered} ^{}m=\frac{\Delta y}{\Delta x} \\ m=\frac{420-285}{6-3} \\ m=\frac{135}{3}=45 \end{gathered}[/tex]Then, by the slope-point form of the line, we can find the equation and then substitute x=0.
[tex]\begin{gathered} y-y_0=m(x-x_0) \\ y-285=45(x-3) \\ y=45x-135+285 \\ y=45x+150 \end{gathered}[/tex]Substituting, x=0.
[tex]\begin{gathered} y=45(0)+150 \\ y=150 \end{gathered}[/tex]At the beginning of the collection, he has 150 cards.
A is the set of even numbers greater than or equal to 4 and less than or equal to 8B=1-29, -25,-24, -22, -21, 22, 27(a) Find the cardinalities of A and B.n(A)=3(b) Select true or false.12 € A22 € B67 A-24 € BTruen(B) = 1FalseX%S3
Set A is composed of all the even numbers equal or greater than 4 and equal or less than 8 so set A is:
[tex]A=\lbrace4,6,8\rbrace[/tex]The cardinalities of A and B are equal to their number of elements so we have n(A)=3 and n(B)=7.
With both sets explicitly written we can complete the true or false table. The only thing to take into account is that the symbol ∈ means "belongs to" and that ∉ means "does not belong to".
The first statement of the table is:
[tex]12\in A[/tex]This is false because 12 does not belong to set A since it is not included in it.
The second statement is:
[tex]22\in B[/tex]As you can see 22 is in deed one of the elements of set B which means that this statement is true.
The third one is:
[tex]6\notin A[/tex]This statement is false because as we saw before 6 is an element of set A.
The last statement is:
[tex]-24\in B[/tex]As you can see -24 is one of the elements of set B so this statement is true.
AnswersFalse
True
False
True
angie drew a rectangle. the length of a rectangles she drew is 2 less than three times the width. find the dimensions of the rectangle if the rectangle if the area is 65 square meters
Answer:
The width of the rectangle is 5
The length of the rectangle is 13
Explanation:
Let's call x the length of the rectangle and y the width of the rectangle.
The length is 2 less than 3 times the width, so
x = 3y - 2
And the area is 65 square meters. Since the area is length times width, we get:
xy = 65
Now, we can replace the first equation x = 3y - 2 on the second one to get
(3y - 2)y = 65
3y(y) - 2y = 65
3y² - 2y = 65
3y² - 2y - 65 = 0
So, using the quadratic equation, we get that the solutions to 3y² - 2y - 65 = 0 are
[tex]\begin{gathered} y=\frac{-(-2)\pm\sqrt[]{(-2)^2-4(3)(-65)}_{}}{2(3)} \\ y=\frac{2\pm\sqrt[]{784}}{6} \\ y=\frac{2\pm28}{6} \\ \text{Then} \\ y=\frac{2+28}{6}=\frac{30}{6}=5 \\ or \\ y=\frac{2-28}{6}=\frac{-26}{6}=-\frac{13}{3} \end{gathered}[/tex]The solution is y = 5 because the width can't have a negative length.
Then, replacing y = 5 on the first equation, we get:
x = 3y - 2
x = 3(5) - 2
x = 15 - 2
x = 13
Therefore, the length of the rectangle is 13 meters and the width of the rectangle is 5 meters
given a function's domain values: -9, -6, -3, 0, 1.5, and 3, what is the range of the function's inverse?
Answer:
[tex]-9,-6,-3,0,1.5,3[/tex]Explanation:
Given the domain of the function as;
[tex]-9,-6,-3,0,1.5,3[/tex]Note that the range of the inverse of a function is the same as the domain of a function is the original function.
So the range of the inverse of the function will be;
[tex]-9,-6,-3,0,1.5,3[/tex]A What is the length of side BC of the triangle? 2x+7 4x7 В Enter your answer in the box. 4x units
Solution
For this case we can see from the image given that the value for BC is:
4x units
since that represent the lenght of the side
question will be in picture
Given that the total money Elise had, was $30.
Since she spent $18 on gifts and the rest on 3 corn dogs.
Let the cost of a corn god be 'x' dollars. Then the cost of 3 corn dogs will be,
[tex]\begin{gathered} \text{Expense on corn dogs}=\text{ No. of corn dogs}\times\text{ Cost of 1 corn dog} \\ \text{Expense on corn dogs}=3\times x \\ \text{Expense on corn dogs}=3x \end{gathered}[/tex]So it is found that Elise spent $(3x) on corn dogs.
Since she had spent all the money in buying these two things, the total exprense must be equal to the total money Elise had in the beginning,
[tex]\begin{gathered} \text{Expense on corn dogs}+\text{ Expense on gifts}=30 \\ 3x+18=30 \end{gathered}[/tex]Thus, the required equation is obtained.
Therefore, option (a) is the correct choice.
The sides of a rectangle are in a ratio of 5:7 and the perimeter is 72. Find the area of the rectangle.
Since the sides of the rectangle are in ratio 5: 7
Insert x in the 2 terms of the ratio and find its perimeter using them
[tex]\begin{gathered} L\colon W=5x\colon7x \\ P=2(L+W) \\ P=2(5x+7x) \\ P=2(12x) \\ P=24x \end{gathered}[/tex]Equate 24x by the given perimeter 72 to find the value of x
[tex]24x=72[/tex]Divide both sides by 24
[tex]\begin{gathered} \frac{24x}{24}=\frac{72}{24} \\ x=3 \end{gathered}[/tex]Then the sides of the rectangle are
[tex]\begin{gathered} L=5(3)=15 \\ W=7(3)=21 \end{gathered}[/tex]Since the rule of the area of the rectangle is A = L x W, then
[tex]\begin{gathered} A=15\times21 \\ A=315 \end{gathered}[/tex]The area of the rectangle is 315 square units
Add 28, 362, and 104. Choose the appropriate number for each place value in the sum. Hundreds Tens Ones Intro
ANSWER
[tex]494[/tex]EXPLANATION
We want to add the three numbers:
[tex]28,362,104[/tex]To do this, add the numbers in corresponding place values i.e. units to units, tens to tens, and hundred to hundred:
[tex]\begin{gathered} 28 \\ +362 \\ +104 \\ =494 \end{gathered}[/tex]That is the answer.
Yasmine makes doll clothes for sewing project she used the pattern below to make the front of a skirt for her sister's doll how many square inches of fabric will Yasmine use the skirt pattern
The area of the fabric can be determined as,
[tex]\begin{gathered} A=\frac{1}{2}\times(3.5\text{ in+6 in)}\times4\text{ in} \\ =19in^2 \end{gathered}[/tex]Thus, the required area of the fabric is 19 square inch.
What is the equation of the line perpendicular to the function f(x)=x^2+2x-2 at the point (1,1)?
Equation of a Line
The equation of a line that passes through the point (h, k) and has a slope m, is given by:
[tex]y-k=m(x-h)[/tex]This is known as the point-slope form of the line.
We already know the coordinates of the point (1, 1) but we don't know the value of the slope m. We will find it out by using the rest of the data.
Our line is perpendicular to the function:
[tex]f(x)=x^2+2x-2[/tex]At the given point. To find the slope of the tangent line, we use derivatives:
[tex]f^{\prime}(x)=2x+2[/tex]Substitute x = 1:
[tex]\begin{gathered} f^{\prime}(1)=2\cdot1+2 \\ f^{\prime}(1)=4 \end{gathered}[/tex]Now we know the slope of the tangent line, but our line is perpendicular to that line, so we find the perpendicular slope with the formula:
[tex]\begin{gathered} m_2=-\frac{1}{m} \\ m_2=-\frac{1}{4} \end{gathered}[/tex]We're ready to find the required equation. Substituting the coordinates of the point and the just-found slope:
[tex]y-1=-\frac{1}{4}(x-1)[/tex]This is the point-slope form, but maybe it's required to find the slope-intercept form. Multiply by 4:
[tex]\begin{gathered} 4y-4=-x+1 \\ \text{Add 4:} \\ 4y=-x+5 \\ \text{Divide by 4:} \\ y=-\frac{1}{4}x+\frac{5}{4} \end{gathered}[/tex]This is the answer is slope-intercept form
Evaluate the expression for the given variable values. (p+g) P-29 if p = 4 and q = 8-13-122024
We have the expression:
[tex]\frac{(p+q)^2}{p-2q}[/tex]We have to evaluate it for p = 4 and q = 8.
To do that, we replace the variables p and q with the given values and calculate:
[tex]\begin{gathered} f(p,q)=_{}\frac{(p+q)^2}{p-2q} \\ f(4,8)=\frac{(4+8)^2}{4-2\cdot8} \\ f(4,8)=\frac{12^2}{4-16} \\ f(4,8)=\frac{144}{-12} \\ f(4,8)=-12 \end{gathered}[/tex]Answer: The value of the expression when p = 4 and q = 8 is -12 [Second option]
why is this problem incorrect or correct?
Well, I would like to know more about this problem.
If you are solving an equation it is wrong because you are adding 10 units only in one side of the equation, that is a mistake.
If this is not a process to solve the equation it is correct, it does not matter if you are only adding 10 in one side.
Where C(x) is in hundreds Of dollars. How many bicycles should the shop build to minimize the average cost per bicycle
The average cost is given by the next equation:
[tex]C(x)=0.1x^2-0.5x+5.582[/tex]The graph of the function is
As we can see the parabola opens upwards therefore the minim will be located in the vertex
[tex]x=\frac{-b}{2a}[/tex]in our case
b=-0.5
a=0.1
[tex]x=\frac{0.5}{2(0.1)}=2.5[/tex]2.5 hundred
Then
[tex]2.5\times100=250[/tex]Therefore, the shop should build 250 bicycles