A train travels due north at 40 mph. A car travels toward the train from the same starting point leaving 2 hours later than the train.
If the car is traveling at 60 mph, how many miles will the car have to travel to catch up to the train?
The distance traveled by car has to travel to catch up to the train is 240 miles.
Given that,
A train travels due north at 40 mph.
A car travels toward the train from the same starting point leaving 2 hours later than the train.
If the car is traveling at 60 mph.
We have to determine,
How many miles will the car have to travel to catch up to the train?
According to the question,
The distance will the car have to travel to catch up to the train is determined by using the following formula;
[tex]\rm Speed = \dfrac{Distance}{Time}[/tex]
Let the distance will the car have to travel to catch up to the train be x.
Then,
The distance traveled by car in times t is,
[tex]\rm Speed = \dfrac{Distance}{Time}\\\\Distance = Speed \times Time\\\\ x = 40 \times t\\\\x = 60t[/tex]
And car travels toward the train from the same starting point leaving 2 hours later than the train.
Then,
The distance traveled by train in time (t+2) is,
[tex]\rm Speed = \dfrac{Distance}{Time}\\\\Distance = Speed \times Time\\\\ x = 40 \times( t+2)\\\\x = 40(t+2)[/tex]
Substitute the value of x in equation 2 from equation 1,
[tex]\rm x = 40(t+2)\\\\60t=40t+80\\\\60t-40t=80\\\\20t=80\\\\t = \dfrac{80}{20}\\\\t = 4[/tex]
Therefore,
The distance traveled by car has to travel to catch up to the train is,
[tex]\rm x = 60t\\\\x = 60\times 4\\\\x = 240 \ miles[/tex]
Hence, The distance traveled by car has to travel to catch up to the train is 240 miles.
To know more about System of Equation click the link given below.
https://brainly.com/question/12895249
A. 6.2
B. 0.8
C. 12.3
D. 2.3
Which equation could be used to calculate the sum of the geometric series? 1/4+2/9+4/27+8/81+16/243?
Answer: Sum of the geometric series will be [tex]\frac{763}{972}[/tex]
Step-by-step explanation:
Since we have given that
[tex]\frac{1}{4}+\frac{2}{9}+\frac{4}{27}+\frac{8}{81}+\frac{16}{243}[/tex]
Here,
[tex]a=\frac{2}{9}\\\\r=\frac{a_2}{a_1}\\\\r=\frac{\frac{4}{27}}{\frac{2}{9}}=\frac{4}{27}\times \frac{9}{2}=\frac{2}{3}\\\\n=4[/tex]
As we know the formula for "Sum of n terms in geometric series ":
[tex]S_n=\frac{a(1-r^n)}{1-r}\\S_n=\frac{\frac{2}{9}(1-\frac{2}{3}^4)}{1-\frac{2}{3}}\\S_n=\frac{130}{243}[/tex]
So, Complete sum will be
[tex]\frac{130}{243}+\frac{1}{4}=\frac{520+243}{972}=\frac{763}{972}[/tex]
Hence, Sum of the geometric series will be [tex]\frac{763}{972}[/tex]
What is the center of the circle with the equation x2 + y2 – 10x – 11 = 0 ?
Which shows all the exact solutions of 2sec^2x-tan^4x=-1 ? Give your answer in radians.
You can use the fact that the range of tangent function is whole set of real numbers.
The solutions to the given equation are
[tex]x = \pm \dfrac{\pi}{3 }+ n\pi ; \: n \in \mathbb Z\\[/tex]
[tex]sin^2(\theta) + cos^2(\theta) = 1\\\\1 + tan^2(\theta) = sec^2(\theta)\\\\1 + cot^2(\theta) = csc^2(\theta)[/tex]
It is a fact that tangent ratio has range as all real numbers. We can use this fact along with the second Pythagorean identity to get to the solution of the given equation.
The given equation is [tex]2sec^2x-tan^4x=-1[/tex]
Using the second Pythagorean identity, we get the equation as
[tex]2\sec^2x-\tan^4x=-1\\\\2(1 + \tan^2x) - (\tan^2x)^2= -1\\\\(\tan^2x)^2 -2\tan^2x -3 = 0[/tex]
Assuming [tex]y = tan^2x[/tex], then we get [tex]y \geq 0[/tex]
The equation becomes
[tex](\tan^2x)^2 -2\tan^2x -3 = 0\\\\y^2 - 2y - 3 = 0\\y-3y + y - 3 = 0\\y(y - 3) + 1(y-3) = 0\\(y+1)(y-3) = 0\\y = -1, y = 3[/tex]
As we know that y is non-negative, so only valid solution is y = 3
Thus,
[tex]y = tan^2(x) = 3\\tan(x) = \pm \sqrt{3}\\x = \tan^{-1}(\pm \sqrt{3})[/tex]
Thus,
[tex]x = tan^{-1}(\sqrt{3}) = 60^\circ + n\pi ; \: n \in \mathbb Z\\\\x = tan^{-1}(-\sqrt{3}) = -60^\circ + n\pi ; \: n \in \mathbb Z[/tex]
Thus, the solutions to the given equation are:
[tex]x = \pm 60^\circ + n\pi ; \: n \in \mathbb Z\\[/tex]
Converting to radians,
[tex]x = \pm \dfrac{\pi}{3 }+ n\pi ; \: n \in \mathbb Z\\[/tex]
Thus,The solutions to the given equation are
[tex]x = \pm \dfrac{\pi}{3 }+ n\pi ; \: n \in \mathbb Z\\[/tex]
Learn more about trigonometric ratios here:
https://brainly.com/question/22599614
The exact solutions of the equation 2sec^2x-tan^4x=-1 are x = 2nπ and x = (2n+1)π/2 for integer values of n.
Explanation:The given equation is 2sec^2x-tan^4x=-1.
Let's simplify the equation:
2(1/cos^2x)-(tan^2x)^2 = -1
2/cos^2x - tan^4x = -1
Now, substituting sec^2x = 1/cos^2x and tan^2x = (sinx/cosx)^2, we get:
2(1/cos^2x)-((sinx/cosx)^2)^2 = -1
2/cos^2x - sin^4x/cos^4x = -1
Now, let's substitute sin^2x = 1 - cos^2x:
2/cos^2x - (1-cos^2x)^2/cos^4x = -1
Now, solving for cos^2x:
2/cos^2x - (1-2cos^2x+cos^4x) = -1
2 - 2cos^2x + cos^4x - cos^2x = -cos^2x
cos^4x - 3cos^2x + 2 = 0
Now, we can solve for cos^2x by factoring the quadratic equation:
(cos^2x - 2)(cos^2x - 1) = 0
cos^2x = 2 or cos^2x = 1
Since the range of cos^2x is [0,1], we can discard the solution cos^2x = 2.
Therefore, cos^2x = 1.
Which means, cosx = ±1.
Since the required range is [0,2π], we can take two solutions:
cosx = 1, implies x = 2nπ, where n is an integer.
cosx = -1, implies x = (2n+1)π/2, where n is an integer.
Hence, the exact solutions of the equation 2sec^2x-tan^4x=-1 are x = 2nπ and x = (2n+1)π/2 for integer values of n.
Learn more about trigonometric equations here:https://brainly.com/question/37290767
#SPJ12
Which graph represents the function g(x)=x−1−−−−−√+1 ?
Answer:
First graph
Step-by-step explanation:
Given function,
[tex]g(x)=\sqrt{x-1}+1[/tex]
∵ The point from which the graph of the function passes through will be satisfy the function,
In the first graph,
Graph is passing through (1, 1), (5, 3) and (10, 4),
[tex]1=\sqrt{1-1}+1[/tex]
[tex]3=\sqrt{5-1}+1[/tex]
[tex]4=\sqrt{10-1}+1[/tex]
So, all points (1, 1), (5, 3) and (10, 4) satisfy the function,
In the second graph,
Graph is passing through (-1, -1), (3,1) and (8,2),
Since,
[tex]-1\neq \sqrt{-1-1}+1[/tex]
So, all points of graph 2 do not satisfy the function,
In the third graph,
Graph is passing through (1, -1), (5, 1) and (10, 2),
[tex]-1\neq \sqrt{1-1}+1[/tex]
So, all points of graph 3 do not satisfy the function,
In the fourth graph,
Graph is passing through (-1, 1), (3, 3) and (8, 4),
[tex]1\neq \sqrt{-1-1}+1[/tex]
So, all points of graph 4 do not satisfy the function
A trapezoid with an area of 166.75 in has bases that measure 21 in and 8 in find the height of the trapezoid
The required height of the trapezoid is 11.5 inches.
What is simplification?Simplification involves applying rules of arithmetic and algebra to remove unnecessary terms, factors, or operations from an expression. The goal is to obtain an expression that is easier to work with, manipulate, or solve.
Here,
The formula for the area of a trapezoid is:
A = (b₁ + b₂) * h / 2
where A is the area, b1 and b2 are the lengths of the two bases, and h is the height.
We are given that the area of the trapezoid is 166.75 in² and the lengths of the two bases are 21 in and 8 in. So we can plug these values into the formula and solve for h:
166.75 = (21 + 8) * h / 2
Simplifying the expression on the right side:
166.75 = 29 * h / 2
Multiplying both sides by 2:
333.5 = 29 * h
Dividing both sides by 29:
h = 11.5
So the height of the trapezoid is 11.5 inches.
Learn more about simplification here:
https://brainly.com/question/12501526
#SPJ2
Final answer:
To find the height of a trapezoid given its area (166.75 sq in) and bases (21 in and 8 in), you use the formula-
[tex]A = \frac{1}{2} h(b_1 + b_2)[/tex]
By rearranging to solve for h, you find that the height is 11.5 inches.
Explanation:
The question asks to find the height of a trapezoid given its area and bases. The area of a trapezoid is calculated using the formula A = [tex]\(\frac{1}{2}\)h(b_1 + b_2)[/tex], where A represents the area, h the height, and [tex]b_1[/tex] and [tex]b_2[/tex] the lengths of the two bases. In this case, the area is 166.75 square inches, and the bases measure 21 inches and 8 inches.
To find the height, we can rearrange the formula to-
h = [tex]\(\frac{2A}{b1 + b2}\)[/tex]
Plugging in the given values gives us-
h = [tex]\(\frac{2 \times 166.75}{21 + 8}\)[/tex]
which simplifies to-
h = [tex]\(\frac{333.5}{29}\)[/tex]
Therefore, the height of the trapezoid is 11.5 inches.
Ms.kellsons storage closet is 3ft long,3ft wide,and 7 ft high.can she fit 67 boxes that each have a volume of 1 cubic foot in her closet
Evaluate 8- m/n+p ^2 when m=8 n=2 p=7
Answer: 53.
Step-by-step explanation:
The given expression : [tex]8-\dfrac{m}{n}+p^2[/tex]
To find : The value of the above expression for the values m=8, n=2 and p=7.
For that , we just substitute the gives values m=8, n=2 and p=7 in the above expression by using Substitution property , we get
[tex]8-\dfrac{8}{2}+(7)^2[/tex]
Simplify,
[tex]=8-4+49[/tex]
[tex]=4+49=53[/tex]
Hence, the correct value of the given expression when m=8 n=2 p=7 is 53.
-6(14-7) healppppp plzzzz
You deposit $3000 in an account that pays 5% annual interest. What is the balance after 2 years?
With the deposit of 3000, the balance after 2 years would be $3307.50
How do we find the compound interest for the deposit?
The formula is: A = P(1+r)ⁿ
A is the amount of money accumulated after n years, including interest.
P = $3000
r = 5% = 0.05 (as a decimal)
n = 2 years
A = 3000 × (1+0.05)²
A = 3000 × (1.05)²
A = 3000 × 1.1025
A = 3307.50
Therefore, when a deposit of $3000 is made, the balance after 2 years would be $3307.50
Factor the polynomial. 7x2 + 68xy - 20y2 A) (7x - 2y)(x + 10y) B) (7x - 10y)(x - 2y) C) (7x + 10y)(x + 2y) D) (7x + 2y)(x - 10y)
Simplify the square root of 48 ( this can be expressed as √48)
Suppose you are driving to visit a friend in another state. you are driving 55 miles per hour. you must drive 440 miles total. if you have already driven 275 miles, how long will it take you to reach your destination? use h to represent the number of hours it will take to reach your destination. use the equation 55h + 275 = 440.
Final answer:
To calculate the remaining travel time, subtract the distance already driven from the total distance and divide the result by the speed. It will take 3 hours to reach the destination.
Explanation:
The question involves calculating the remaining travel time to reach a destination, given the current distance traveled, the total distance, and the traveling speed. We're provided with the equation 55h + 275 = 440, where h represents the number of hours needed to reach the destination.
Here's how we solve the equation step-by-step:
Subtract 275 from both sides of the equation to isolate the term with h. This gives us 55h = 440 - 275.
Subtract 275 from 440 to get the remaining distance, which is 165 miles.
Divide the remaining distance by the speed to find the time in hours. So we have h = 165/55.
Calculate the division to find h = 3 hours.
Therefore, it will take 3 hours to reach the destination after having already driven 275 miles at a speed of 55 miles per hour.
If you were to add the length of all the 3/8 pieces which are 4 (number of pieces) what would be the total length
To find the total length of four 3/8-inch pieces, you multiply the length of one piece (3/8 inch) by the number of pieces (4), which equals 1.5 inches.
To calculate the total length of all the 3/8-inch pieces, we multiply the length of one piece by the number of pieces, which is 4. The mathematical expression for this is:
Total length = length of one piece x number of pieces
Total length = 3/8 x 4
To perform the multiplication, multiply the numerators and then the denominators:
Total length = (3 x 4)/(8 x1)
Total length = 12/8 inches
Now, simplify the fraction by dividing the numerator and the denominator by the greatest common divisor, which is 4:
Total length = (12/4)/(8/4)
Total length = 3/2 inches
And since 3/2 inches is equal to 1.5 inches, the total length of all four pieces is 1.5 inches.
A puppy and a kitten are 180 meters apart when they see each other. The puppy can run at a speed of 25 m/sec, while the kitten can run at a speed of 20 m/sec.
a
How soon will they meet if they simultaneously start running towards each other?
4 seconds
Let they meet after time 't' seconds
Let distance covered by puppy in 't' seconds be 'x' meters
So distance covered by kitten in 't' seconds is (180 - x ) meters
Time = Distance / Speed
so,
t = x / 25 for puppy
t = (180-x) / 20 for kitten
On solving,
x = 100 meters
t = 4 seconds
Find the value of y that makes the equation y = -2x + 4 true when x = 1. Select one: a. 2 b. -2 c. 6 d. -6
Alexis and tasha challenge each other to a typing test. Alex typed 54 words in one minute which was 123% of what tasha typed. how many words did tasha type in one mintue
Find the width of a rectangular patio with a length of 16 ft and an area of 200 square feet
Ajuda eu !!!!!!!!!!!!!!!!
Suppose that an individual has a body fat percentage of 12.8% and weighs 131 pounds. how many pounds of his weight is made up of fat? round your answer to the nearest tenth.
The probability of event A is .43, the probability of event B is .32, and the probability of event C is .66. What is the probability of all three events occurring at the same time?
Final answer:
The probability of independent events A, B, and C all occurring together is found by multiplying their individual probabilities: P(A AND B AND C) = P(A) × P(B) × P(C) = 0.090528.
Explanation:
The question involves calculating the probability of all three independent events happening simultaneously. To find the combined probability of independent events A, B, and C occurring at the same time, we use the multiplication rule of probability.
This rule states that if the events are independent, the probability of all events occurring is the product of their individual probabilities.
Here are the probabilities for each event:
Probability of event A (P(A)) = 0.43
Probability of event B (P(B)) = 0.32
Probability of event C (P(C)) = 0.66
The probability of events A, B, and C all occurring together is:
P(A AND B AND C) = P(A) × P(B) × P(C) = (0.43) × (0.32) × (0.66) = 0.090528
Tell whether each set of ordered pairs representd a function. Justify your answers.
a. (0,0), (1,1), (2,2), (3,3), (4,4)
b. (0,8), (1,6), (2,4), (3,2), (4,0)
c. (3,0), (3,1), (3,2), (3,3), (3,4)
Algebra question ( Matrices and Determinants ) 20 points
A poll was conducted in states A, B, and C to determine which candidate voters would most likely vote for. The results are displayed in the table below:
What is P(conservative | State B)?
State A State B State C Total
Liberal 17 20 12 49
Conservative 23 15 13 51
Total 40 35 25 100
Answer:
15/35 = 3/7 = 0.428571
Explanation:
We are asked to find the probability that a given voter is conservative given they are in State B.
Going to the State B column, we see that there are 15 conservatives. This is out of a total of 35 voters in state B; this makes the probability 15/35.
Probability helps us to know the chances of an event occurring. The probability that a voter from State B is supporting Conservative is 0.4285.
What is Probability?Probability helps us to know the chances of an event occurring.
[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
We're supposed to figure out how likely it is that a specific voter in State B is a conservative.
There are 15 conservatives in State B, according to the column. This is based on a total of 35 voters in state B, resulting in a 15/35 probability. Therefore,
P(conservative | State B) = 15/35 = 3/7 = 0.4285
Hence, the probability that a voter from State B is supporting Conservative is 0.4285.
Learn more about Probability:
https://brainly.com/question/795909
#SPJ2
A varies jointly as b and c. Find when b = 7 and c = 9, if the constant is the variation is 3.
A) 189
B) 7/3
C) 21
Answer:
Option A is correct
Value of A is, 189
Step-by-step explanation:
Joint variation states:
A varies jointly as b and c.
⇒[tex]A \propto b[/tex] and [tex]A \propto c[/tex]
⇒[tex]A \propto bc[/tex]
then the equation is in the form of:
[tex]A = k \cdot bc[/tex]
where, k is the constant of variation.
As per the statement:
A varies jointly as b and c.
⇒[tex]A = k \cdot bc[/tex] ....[1]
To find A .
Substitute the given values b = 7 , c = 9 and k = 3 in [1]
then;
[tex]A = 3 \cdot 7 \cdot 9 = 21 \cdot 9 = 189[/tex]
Therefore, the value of A is, 189
You draw a single card from a standard deck of cards. what is the probability of drawing a diamond?
In a weekly lottery, ten ping-pong balls numbered 0 to 9 are placed in each of six containers, and one ping-pong ball is drawn from each container. To win the prize, a participant must correctly identify the ping-pong ball that is drawn from each of the six containers. If Juan played the lottery last week and didn’t win, what is the probability that he will win this week?
The probability that he will win this week is 0.0001%.
What is Probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.
The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
The degree to which something is likely to happen is basically what probability means.
Given:
Ten ping-pong balls numbered 0 to 9 are placed in each of six containers.
It will not effect the outcome if Juan didn't win last week.
So, the probability that he will win this week
= [tex](1/10)^6[/tex]
= 1/1,000,000
= 0.000001
= 0.0001%
Hence, the probability that he will win this week is 0.0001%.
Learn more about probability here:
https://brainly.com/question/11234923
#SPJ2
How does the volume of an oblique cylinder change if the radius is reduced to 2/9 of its original size and the height is quadrupled?
A. V=2/81pir^2h
B. V=16/81pir^2h
C. V=4/9pir^2h
D. V=16/9pir^2h
sin^{2} theta +cos theta = 2
Use the pythagorem identity sin^{2} theta +cos^{2} theta =1 to replace sin^{2} theta in the given equation
Applying Pythagorean Identity, the original equation sin² theta +cos theta = 2 can be altered by substituting sin² theta with 1 - cos² theta. This forms a quadratic equation: cos² theta - cos theta - 1 = 0, which can be solved using the quadratic formula to find the possible values of cos theta.
Explanation:To solve the equation sin² theta +cos theta = 2 by using the Pythagorean identity sin² theta +cos² theta =1, we can substitute sin² theta in the original equation with 1 - cos² theta.
Step-by-step solution:
We start by replacing sin² theta in the equation with 1 - cos² theta. So, the equation becomes 1 - cos² theta + cos theta - 2 = 0.This can be rearranged to form a quadratic equation as cos² theta - cos theta - 1 = 0.We can solve this quadratic equation using the quadratic formula cos theta = [- (-1) ± √{(-1)²-4(1)(-1)} ] /2(1).Computing the values, we get the two possible values of cos theta.Learn more about Pythagorean Identity here:https://brainly.com/question/24287773
#SPJ2