Consider the graph of f(x) = 5 ^ x + 1 1. Explain how to find the average rate of change between x = 0 and x = 4 . What is the average rate of change ?
Answers
1. To determine the average rate of change of a function "f(x)" between the points "x = a" and "x = b" we use the following formula:
[tex]A=\frac{f(b)-f(a)}{b-a}[/tex]2. In this case, we have the following function:
[tex]f(x)=5^x+1[/tex]And we have the points:
[tex]\begin{gathered} x=0 \\ x=4 \end{gathered}[/tex]Now we determine the value of f(b) by replacing x = 4 in the function:
[tex]\begin{gathered} f(4)=5^4+1 \\ f(4)=626 \end{gathered}[/tex]Now we determine f(0):
[tex]\begin{gathered} f(0)=5^0+1 \\ f(0)=1+1=2 \end{gathered}[/tex]Replacing in the formula for the average rate of change we get:
[tex]A=\frac{626-2}{4-0}[/tex]Solving the operations:
[tex]A=\frac{624}{4}=156[/tex]Therefore, the average rate of change is 156.
Related Questions
Write (3-2i)^3 in simplest a + bi form.
Answers
SOLUTION
We want to write
[tex]\begin{gathered} \mleft(3-2i\mright)^3\text{ in simplest form } \\ a+bi \end{gathered}[/tex]This means we have to expand
[tex](3-2i)^3[/tex]Applying perfect cube formula, we have
[tex]\begin{gathered} \mleft(a-b\mright)^3=a^3-3a^2b+3ab^2-b^3 \\ \text{where } \\ a=3,\: \: b=2i \end{gathered}[/tex]We have
[tex]\begin{gathered} (a-b)^3=a^3-3a^2b+3ab^2-b^3 \\ \mleft(3-2i\mright)^3=3^3-(3\times3^2\times2i)+(3\times3\times(2i)^2)-(2i)^3_{} \\ =27-(27\times2i)+(9\times(2i)^2)-(2i)^3_{} \end{gathered}[/tex]This becomes
[tex]\begin{gathered} \text{note that i = }\sqrt[]{-1} \\ i^2=\sqrt[]{-1^2}=-1 \\ So\text{ we have } \\ =27-(27\times2i)+(9\times(2i)^2)-(2i)^3_{} \\ 27-54i+(9\times4i^2)-(8i^2\times i) \\ 27-54i+(9\times4\times-1)-(8\times-1\times i) \\ 27-54i-36+8i \\ -9-46i \end{gathered}[/tex]Hence the answer is
[tex]-9-46i[/tex]in the diagram, AB=9, DB=5, and BC=12. if m< B = 90, what is the perimeter of ADC ?
Answers
To answer this question, we can draw the triangle as follows:
We can start answering this question, finding the value of the side AC (one of the sides of the triangle ADC (not shown)). Then, we can apply the Pythagorean Theorem to find AC (the hypotenuse in this case). Thus, we have:
[tex]AC^2=AB^2+BC^2\Rightarrow AC^2=(9)^2+(12)^2=81+144[/tex]Then, we have:
[tex]AC^2=225\Rightarrow\sqrt[]{AC^2}=\sqrt[]{225}\Rightarrow AC=15[/tex]Then, the measure of the side AC = 15.
Now, we need to find the measure of the side DC. We can also need to apply the Pythagorean Theorem to find it:
We have that DB = 5, and BC = 12. Then:
[tex]DC^2=DB^2+BC^2=5^2+12^2=25+144\Rightarrow DC^2=169[/tex]Now, we need to take the square root to both sides of the expression to solve for DC:
[tex]\sqrt[]{DC^2}=\sqrt[]{169}\Rightarrow DC=13[/tex]To find the perimeter of the triangle ADC, we have:
1. AD = AB - DB ---> AD = 9 - 5 ---> AD = 4 (we deduce this from the given data in the question.)
2. AC = 15 (found in the first step using the Pythagorean Theorem.)
3. DC = 13 ((found in the second step using the Pythagorean Theorem.)
Therefore, the perimeter of the triangle ADC is the sum of all of its sides, then, we have:
[tex]P_{\text{trangleADC}}=AD+AC+DC\Rightarrow P_{triangleADC}=4+15+13=32[/tex]Therefore, the perimeter of the triangle ADC is equal to 32 units.
(We can apply the Pythagorean Theorem twice since we have a right triangle in both cases: triangle ABC and triangle DBC.)
Solve in inequality 0.2 (30 + x). - 0.3 (30-x) >2
Answers
The inequality 0.2 (30 + x) - 0.3 (30 - x) >2 is solved to get x > 10
How to solve the inequalityThe given inequality is 0.2 (30 + x) - 0.3 (30 - x) >2
0.2 (30 + x) - 0.3 (30 - x) >2
expanding the parenthesis
6 + 0.2x - 9 + 0.3x > 2
collecting like terms
6 - 9 + 0.2x + 0.3x > 2
-3 + 0.5x > 2
0.5x > 2 + 3
0.5x > 5
dividing by the coefficient of x which is 0.5
x > 10
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an average bath use 35 liters if water,while a five minutes shower only uses 12.5 liters of water.how many milliliters are you saving if you take a five minutes shower instead of a bath
Answers
To find the saving of water you have to substract the average bath to the average shower
[tex]35-12.5=22.5[/tex]So you save 22.5 liters of water.
Now we have to convert the liters to mililiters, this can be done if we rememeber that a liter contains 1000 mililiters, then we multiply our result by 1000.
[tex]22.5(1000)=22500[/tex]Therefore we save 22500 mililiters of water.
Tyler has already taken 35 credit hours and plans on taking 15 hours per semester until he graduates. Does this describe a linear or exponential function?
Answers
Given:
Tyler has already taken 35 credit hours.
He plans on taking 15 hours per semester.
Let the number of semesters = x
So, the function that describes this will be:
[tex]y=15x+35[/tex]So, the function represents a line.
So, the answer will be a Linear function.
f(x) = 2.5x - 10.5if X= 2
Answers
Answer:
-5.5
Explanation:
Gven the below function;
[tex]f(x)=2.5x-10.5[/tex]when x = 2, it means we should substitute x = 2 into the above function or find f(2);
[tex]\begin{gathered} f(2)=2.5(2)-10.5 \\ =5-10.5 \\ =-5.5 \\ \therefore f(2)=-5.5 \end{gathered}[/tex]If you select one card at random from a standard deck of 52 cards, what is the probability that the card is black OR a 6?
Answers
Since there are 52 cards in the standard deck
Since half of them are in black
Then the probability of getting a black card is
[tex]\begin{gathered} P(b)=\frac{\frac{52}{2}}{52} \\ P(b)=\frac{26}{52} \end{gathered}[/tex]Since there are 4 cards of 6, then
The probability of getting 6 is
[tex]P(6)=\frac{4}{52}[/tex]OR in probability means adding, then
The probability of getting a black card or a 6 is
[tex]\begin{gathered} P(b\text{ or 6)=}\frac{26}{52}+\frac{4}{52} \\ P(b\text{ or 6) =}\frac{30}{52} \end{gathered}[/tex]We can simplify it by dividing up and down by 2
[tex]\begin{gathered} P(b\text{ or 6)=}\frac{\frac{30}{2}}{\frac{52}{2}} \\ P(b\text{ or 6)=}\frac{15}{26} \end{gathered}[/tex]The answer is P(b or 6) = 30/52 OR 15/26
How much do I need to increase a radius of a circle to increase it's area 10 times?
Answers
Given that
It is said that we have to find the amount by which the radius will be increased such that the area is increased by 10 times.
Explanation -
The formula for the area of the circle is given as
[tex]\begin{gathered} Area=\pi\times r^2 \\ \\ A=\pi r^2-----------(i) \\ \\ where\text{ r is the radius of the circle.} \end{gathered}[/tex]Now the new area is 10 times the previous one.
Let the new area be A' and the new radius be R.
Then,
[tex]\begin{gathered} A^{\prime}=\pi\times R^2 \\ \\ As\text{ A'=10}\times A \\ \\ Then\text{ substituting the value of A' we have} \\ \\ 10\times A=\pi\times R^2 \end{gathered}[/tex]Now again substituting the value of A we have
[tex]\begin{gathered} \pi\times R^2=10\times\pi\times r^2 \\ \\ R^2=10r^2 \\ \\ R=\sqrt{10}\times r \end{gathered}[/tex]Hence the new radius will be √10 times the initial radius such that the area gets increased by 10 times.
Final answer - Therefore the final answer is √10 times.
Rachel is bowling with her friends. Her bowling ball has a radius of 4.1 inches. As she bowls she tracks the location of the finger hole above the ground. She starts tracking the location when the finger hole is at the 12 o'clock position and she notices that she got some backspin on the ball and it rotates counter-clockwise.Write a function f that determines the height of the finger hole above the ground (in inches) in terms of the number of radians a the ball has rotated since she started tracking the finger hole. (Note that aa is a number of radians swept out from the 12-o'clock position.)f(a)=
Answers
Given that radius is r= 4.1 inches.
let track the location of finger hole is at 12 o'clock.
i.e. the angle is 0 degree.
at 12 o'clock
[tex]\theta=0[/tex]Now when the finger hole changed by 45 degree:
[tex]\theta=45[/tex]Now convert 45 degree into radians:
[tex]\begin{gathered} \theta=45\times\frac{\pi}{180} \\ \theta=\frac{\pi}{4} \end{gathered}[/tex]So angle is such that:
[tex]\begin{gathered} \theta\in\lbrack0,\frac{\pi}{4}\rbrack \\ 0\leq\theta\leq\frac{\pi}{4} \end{gathered}[/tex]Now calculate the measure of function in polar coordinates:
[tex]\begin{gathered} \theta=0,\text{ f(}\theta\text{)}=r \\ \theta=\frac{\pi}{2},\text{ f(}\theta)=r\cos \theta \end{gathered}[/tex]Taking measurement of function:
[tex]\begin{gathered} f(\theta)=r+r\cos \theta \\ f(\theta)=r(1+\cos \theta) \end{gathered}[/tex]So the function become and the limit is:
[tex]f(\theta)=r(1+\cos \theta),\text{ 0}\leq\theta\leq\frac{\pi}{4}[/tex]What are the Characteristics for rhombus
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The characteristics of a rhombus are the following:
• Opposite sides are parallel ang congruent (equal)
,• The diagonal lines are bisectors (they cut in half) the internal angles
,• The two diagonals have different legths (usually represented by d and D)
,• The point where the two diagonals meet is the center of the rhombus
The illumination provided by a car's headlight varies inversely as the square of the distance from the headlight. A car's headlight produces an illumination of 3.75 footcandles at a distance of 40 feet. What is the illumination when the distance is 50 feet?
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The illumination should be represented by y, while the distance from the headlight should be represented by x, therefore the inverse relationship between both variables is shown as;
[tex]\begin{gathered} y=\frac{k}{x^2} \\ \text{When y=3.75, then x=40. Therefore;} \\ 3.75=\frac{k}{40^2} \\ 3.75=\frac{k}{1600} \\ 3.75\times1600=k \\ k=6000 \\ \text{Hence, when the distance (x) is 50 feet} \\ y=\frac{k}{x^2} \\ y=\frac{6000}{50^2} \\ y=\frac{6000}{2500} \\ y=2.4 \end{gathered}[/tex]The illumination at a distance of 50 feet is therefore 2.4 footcandles
Simplify the square root:square root of negative 72 end rootAnswer choices Include:2 i square root of 186 i square root of 218 i square root of 22 i square root of 6
Answers
We need to simplify the next square root:
[tex]\sqrt[]{-72}[/tex]First, we need to rewrite the expression as:
[tex]\sqrt{-72}=\sqrt[]{-1}\ast\sqrt[]{72}[/tex]Where √-1 = i
Therefore:
[tex]\sqrt{-1}\ast\sqrt{72}=\sqrt[]{72}\text{ i}[/tex]Finally, we can simplify inside of the square root:
[tex]\sqrt[]{72}i=\sqrt{6\ast6\ast2}i=\sqrt{6^2\ast2}i=6i\sqrt[]{2}^[/tex]Therefore, the correct answer is "6 i square root of 2".
Find the lengths of the diagonals of rectangle WXY Z where WY-2x + 34 and XZ = 3x – 26The length of each diagonal isunits.
Answers
To solve the exercise, you can first draw a picture to better understand the statement. So,
Now, in a rectangle, the lengths of the diagonals measure the same. So,
[tex]\begin{gathered} WY=XZ \\ -2x+34=3x-26 \end{gathered}[/tex]To solve for x first subtract 34 from both sides of the equation
[tex]\begin{gathered} -2x+34-34=3x-26-34 \\ -2x=3x-60 \end{gathered}[/tex]Subtract 3x from both sides of the equation
[tex]\begin{gathered} -2x-3x=3x-60-3x \\ -5x=-60 \end{gathered}[/tex]Divide by -5 into both sides of the equation
[tex]\begin{gathered} \frac{-5x}{-5}=\frac{-60}{-5} \\ x=12 \end{gathered}[/tex]Finally, replace the value of x in the length of any of the diagonals, for example, the diagonal WY
[tex]\begin{gathered} WY=-2x+34 \\ WY=-2(12)+34 \\ WY=-24+34 \\ WY=10 \end{gathered}[/tex]Therefore, the length of each diagonal is 10 units.
Determine the point estimate of the population mean and margin of error for the confidence interval.
Lower bound is 17, upper bound is 29.
The point estimate of the population mean is
The margin of error for the confidence interval is
...
Answers
The point estimate of the population mean is 23 and the margin of error for the confidence interval is 6.
In the given question, we have to find the value of the point estimate of the population mean and the margin of error for the confidence interval.
From the given question,
Lower bound is 17.
Upper bound is 29.
So the point estimate of the population mean is
Point Estimate = (Lower Bond+Upper Bond)/2
Point Estimate = (17+29)/2
Point Estimate = 46/2
Point Estimate = 23
Now finding the margin of error for the confidence interval.
Margin of error = Upper Bound-Point Estimate
Margin of error = 29-23
Margin of error = 6
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Convert the radian measure to degreemeasure. Then, calculate the arc length thatcorresponds to a circle with a 35-centimeterdiameter. Round your answer to the nearesttenth.
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We will have the following:
First,:
[tex]\frac{4\pi}{15}=\frac{4\pi}{15}\ast\frac{180}{\pi}=48[/tex]Then, the arc length will be:
[tex]\begin{gathered} s=(\frac{15}{2})(\frac{4\pi}{15})\Rightarrow s=2\pi \\ \\ \Rightarrow s\approx6.3 \end{gathered}[/tex]So, the arc length is approximately 6.3 cm.
2. Write the equation of g(x) given the table below. X -2 --1 1 g(x) 3 6 12
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let us first calculate the slope:
[tex]m=\frac{6-3}{-1--2}=\frac{3}{1}=3[/tex]having the slope, we can use the slope-point equation and we get that
[tex]\begin{gathered} y-12=3(x-1)=3x-3 \\ y=3x-3+12=3x+9 \end{gathered}[/tex]so g(x)=3x+9
In January, it snowed 36.45 inches. In December it snowed 19.7 inches. How many more inches did it snow in January than in December?
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Determine the difference in height of snow.
[tex]\begin{gathered} h=36.45-19.7 \\ =16.75 \end{gathered}[/tex]Thus, 16.75 inches more snowed in
How does linear inequality work?
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To graph a inequality we first need to grapg the equality, in this case the line y=3-1/3x, and then we check which of the two sets of points fullfil the inequality and paint this set. in this case the inequality is estrict so it doesn't include the boundary line.
4. Write an equation for a line that isperpendicular to y = -4.
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The equation y = -4 is a horizontal line at the point -4 in the y-axis.
In order to find a perpendicular line to this equation, we can choose any vertical line in the form:
[tex]x=a[/tex]Where 'a' is any real constant.
So an equation for a line perpendicular to y = -4 would be:
[tex]x=2[/tex]17. What is the value of x in the rhombusbelow?AC(x+40)B3x"D
Answers
Remember that
In a rhombus, diagonals bisect each other at right angles (perpendicular)
so
that means
(x+40)+(3x) =90 degrees ---------> by complementary angles
solve for x
4x+40=90
4x=90-40
4x=50
x=12.5determine whether the graph of the given equations are parallel, perpendicular, or neither.y=2x+15y= -2x+3
Answers
1. When two lines have the same slope they are parallel.
2 . When the quotient of two lines is -1 they are perpendicular.
The slope of
y = 2x + 15 is 2
the slope of
y = -2x + 3 is -2
since -2/2 = -1 they are perpendicular :)
ine temperaturtemperature at midnight?-53Allie scores 4 points in the first round of a card game. In the next round, she loses 6 points. Then she scores 4more points. How many points does Allie have after three rounds?on
Answers
3
Allie scores 4 points in the first round of a card game. In the next round, she loses 6 points. Then she scores 4.
How many points does Allie have after three rounds?
First round: 4 points (positive)
Second round = lose 6 points (negative)
Third round = scores 4 (positive )
Add and subtract all the points
4-6+4 = 2
Use trig ratios to find the missing side of the triangle below. Show all of your work. Round your answer to the nearest 10th.
Answers
Answer:
[tex]x\approx12.7[/tex]Step-by-step explanation:
To solve this situation, we can use trigonometric ratios or relationships to find x. Trigonometric ratios are represented by the following diagram and formulas:
Therefore, to find x, use the sen relationship:
[tex]\begin{gathered} \sin (65)=\frac{x}{14} \\ x=14\cdot\sin (65) \\ x=12.688 \\ \text{ Rounding to the nearest 10th:} \\ x\approx12.7 \end{gathered}[/tex]Which points are separated by a distance of 6 units? O A. (2, 4) (2, 2) B. (1,8) (1,2) c. (3, 1) (3,6) D. (5,6) (5,5)
Answers
Recall that the distance formula is given by
[tex]d=\sqrt[]{\mleft({x_2-x_1}\mright)^2+\mleft({y_2-y_1}\mright)^2}[/tex]We are asked to find out which of the given points have a distance of 6 units?
Let us analyze each option.
A. (2, 4) (2, 2)
[tex]\begin{gathered} d=\sqrt[]{({x_2-x_1})^2+({y_2-y_1})^2} \\ d=\sqrt[]{({2-2_{}})^2+({2-4})^2} \\ d=\sqrt[]{({0})^2+({-2})^2} \\ d=\sqrt[]{4}^{} \\ d=2 \end{gathered}[/tex]Option A does not have a distance of 6 units, so it is not the correct option.
B. (1,8) (1,2)
[tex]\begin{gathered} d=\sqrt[]{({x_2-x_1})^2+({y_2-y_1})^2} \\ d=\sqrt[]{({1-1})^2+({2-8})^2} \\ d=\sqrt[]{({0})^2+({-6})^2} \\ d=\sqrt[]{36}^{} \\ d=6 \end{gathered}[/tex]As you can see, the distance between these points is exactly 6 units.
Therefore, Option B is the correct answer.
c. (3, 1) (3,6)
[tex]\begin{gathered} d=\sqrt[]{({x_2-x_1})^2+({y_2-y_1})^2} \\ d=\sqrt[]{({3-3})^2+({6-1})^2} \\ d=\sqrt[]{(0)^2+({5})^2} \\ d=\sqrt[]{25} \\ d=5 \end{gathered}[/tex]Option C does not have a distance of 6 units, so it is not the correct option.
D. (5,6) (5,5)
[tex]\begin{gathered} d=\sqrt[]{({x_2-x_1})^2+({y_2-y_1})^2} \\ d=\sqrt[]{({5-5})^2+({5-6})^2} \\ d=\sqrt[]{({0})^2+({-1})^2} \\ d=\sqrt[]{1}^{} \\ d=1 \end{gathered}[/tex]Option D does not have a distance of 6 units, so it is not the correct option.
what is a division expression with a quotient that is greater than 8 divided by 0.001
Answers
SOLUTION
Write out the given expression
[tex]\frac{8}{0.001}[/tex]An expression with a quotient that is greater than the expression above is to increase the numerator and leave the denominator unchanged
Hence, we have
[tex]\frac{9}{0.001}is\text{ greater than }\frac{8}{0.001}[/tex]Therefore
The division expression with a quotient that is greater than 8 divided by 0.001 is
9/0.001
Which values of x would make a polynomial equal to zero if the factors of thepolynomial were (x+6) and (x+9)?
Answers
Given
(x+6) and (x+9) are the factors of a polynomial.
To find: Which values of x would make a polynomial equal to zero?
Explanation:
It is given that,
(x+6) and (x+9) are the factors of a polynomial.
Then, the polynomial can be written as,
[tex]p(x)=(x+6)(x+9)[/tex]Also, if (x+a) is a factor of a polynomial p(x).
Then, p(-a)=0.
Therefore,
For the factors (x+6) and (x+9),
The polynomial p(x) is zero at x=-6, and x=-9.
Hence, the answer is x = -6, -9.
an 20. Dequan spent 44 minutes yesterday cleaning 3 bathrooms and the kitchen at home. He spent the same amount of time cleaning each place, and then 8 minutes putting all the supplies away after. Write an algebraic equation and then solve to find out how many minutes m he spent on each room.
Answers
Let:
b = minutes spent cleaning the 3 bathrooms
k = minutes spend cleaning the kitchen
He spent the same amount of time cleaning each place, therefore:
k = b
He spent 44 minutes in total, besides he spent 8 minutes putting all the supplies away after:
b + k + 8 = 44
since k = b = m:
m + m + 8 = 44
2m + 8 = 44
Solving for m:
2m = 44 - 8
2m = 36
m = 36/2
m = 18
Therefore, he spent 18 minutes in each room
.The balance on Mr. Finch's credit card is -$210. It is 3 times the balance on Mr. Nguyen's credit card. Find the quotient -210 ÷ 3 and explain what it means in this context.
Answers
Given: Balance of Finch's card is = -$210.
This is 3 times the balance on Mr. Nguyen's credit card.
To find: -210/3.
Explanation:
Let the balance on Mr. Nguyn's card be = x.
The balance of Mr. Finch's card is 3 times Mr. Nguyen's card.
Mathematically this can be expressed as:
[tex]-210=3x[/tex]Now, the value of x or "Mr. Nguyen's credit card balance" can be calculated as:
[tex]\begin{gathered} x=\frac{-210}{3} \\ x=-70 \end{gathered}[/tex]Therefore, the term -210/3 represents Mr. Nguyen's credit card balance and its value is -$70.
Final Answer: The term -210/3 represents Mr. Nguyen's credit card balance and its value is -$70.
Sandra has a bag of animal cookies. The bag contains the cookies below. What is the probability that Sandra chooses a bear cookie first, eats it, and thenselects a lion cookie?9 lions5 elephants 3 tigers9 Bears 18/5218/5181/65081/676
Answers
ANSWER:
The probability of choosing a bear first then a lion is 81/650
SOLUTION:
This is a permutation and probability problem
The total cookies are 26
The combination for the total cookies is 26 * (26-1) = 26*25
The permutation for choosing a bear then a lion is 9 * 9
The probability is the permutation of choosing the bear over the permutation in total combination for total cookies
[tex]\frac{9\times9}{26\times25}=\frac{81}{650}[/tex]In the diagram below of parallelogram ROCK,mZC is 70° and mZROS is 65º.Oс70%650RSK.What is mZKSO?
Answers
Answer:
∠KSO = 135°
Explanation:
On a parallelogram, the opposite angles have the same measure. It means that the measure of ∠R is:
∠R = ∠C
∠R = 70°
On the other hand, the sum of the interior angles of a triangle is equal to 180°, so we can calculate the measure of ∠RSO as:
∠RSO = 180 - ∠ROS - ∠R
∠RSO = 180 - 65 - 70
∠RSO = 45°
Because the angles RSO, ROS, and R form the triangle ROS.
Finally, ∠RSO and ∠KSO form a straight line, so their sum is equal to 180°. Then, we can calculate ∠KSO as:
∠KSO = 180 - ∠RSO
∠KSO = 180 - 45
∠KSO = 135°
Then, the answer is 135°