Equivalent expression are expressions that have equal values, when expanded. The equivalent expression of ln(2x)^4 is 4ln(2) + 4ln(x))
How to determine the expanded expressionThe expression is given as:
[tex]\ln(2x)^4[/tex]
Apply the following logarithmic rule to the above equation
[tex]\ln(a)^b = b\ln(a)[/tex]
So, we have:
ln(2x)^4 = 4ln(2x)
Next, apply the following product rule of logarithm to the above equation
ln(ab) = ln(a) + ln(b)
So, we have:
ln(2x)^4 = 4 * [ln(2) + ln(x)]
Expand the bracket
ln(2x)^4 = 4ln(2) + 4ln(x))
Hence, the equivalent expression of ln(2x)^4 is 4ln(2) + 4ln(x))
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Answer:
ln (2x)4
✅ 4 ln 2 + 4 ln x
❎ 4 ln 2 + ln x
❎ 8 ln x
ln
4y5
x2
❎ ln 4 - 2 ln x - 5 ln y
✅ ln 4 - 2 ln x + 5 ln y
❎ -8 ln x + 5 ln y
Help me :((
What rule describes the rotation about the origin?
(x,y) →
How many degrees was the figure rotated about the
origin?
Answer: 1. C
2. B
Step-by-step explanation:)
The rotation rule describes the rotation about the origin in mathematics. The new coordinates of a point (x, y) after rotation can be determined using trigonometric functions. The angle of rotation can be calculated using the arc length formula.
Explanation:In mathematics, the rule that describes rotation about the origin is referred to as the rotation rule or the radian rule. When a point (x, y) is rotated about the origin, the new coordinates (x', y') can be determined using the following formulas:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
The angle of rotation can be calculated using the arc length formula:
Angle = (Arc Length / Circle Circumference) * 360 degrees
For example, if the arc length separating the two hands of a clock is 1.0 meter and the radius of the clock is 0.70 meters, the angle of rotation can be calculated as follows:
Angle = (1.0 / (2 * pi * 0.70)) * 360 = 81 degrees
What is the solution set for the following graph below?
The solution set is where the black dot is located, which is on the number 5.
The solution would be 5
In the formula d = StartRoot (x 2 minus x 1) squared + (y 2 minus y 1) squared EndRoot, how does each subtraction expression relate to the Pythagorean theorem? Each subtraction expression represents the length of one leg of a right triangle with a hypotenuse of length d. Each subtraction expression represents the length of the hypotenuse of a right triangle with a leg of length d. Each subtraction expression represents half the length of the hypotenuse of a right triangle with a hypotenuse of length d. Each subtraction expression represents the square of the length of the hypotenuse of a right triangle with a hypotenuse of length d.
Answer:
Each subtraction expression represents the length of one leg of a right triangle with a hypotenuse of length
Step-by-step explanation:
Each subtraction expression x2 - x1 or y2 - y1 represents the length of a side of a right angled triangle. These length of sides are then used to calculate the resulting hypotenuse side using the Pythagorean theorem. In this case, the resulting hypotenuse is the length calculated.
Answer:
the answer is A)Each subtraction expression represents the length of one leg of a right triangle with a hypotenuse of length.
Step-by-step explanation:
took it on edge
Between the two investment plans listed below, which will have the greatest future value and by what amount? Round all answers to the nearest cent.
A 2-column table with 3 rows. Column 1 is labeled 401 (k) with entries An employee contribution of 9 percent on an annual salary of 45,624 dollars and Employer matches 3 percent of employee contribution, compounded annually at 1.2 percent, annual contributions for 30 years. Column 2 is labeled Roth I R A with entries monthly deposit of 352 dollars and 45 cents, compounded monthly at 1.2 percent, monthly contributions for 30 years.
Just copy this: To determine the future value for the 401(k), we need to determine the amount contributed annually. It is stated that the amount the employee contributes to the fund is 9% of $45,624. $45,624(0.09) = $4,106.16 The employer contributes a maximum of 3% of the employee contribution. Therefore: $45,624(0.03) = $1368.72. Therefore, the total annual contribution is $5,474.88, giving a future value of $196,302.40. For the Roth IRA, the monthly contributions are $352.45 giving a future value of $152,636.09. Therefore, the 401(k) has a greater future value by $43,666.31.
Answer:
from original question:
Step-by-step explanation:
To determine the future value for the 401(k), we need to determine the amount contributed annually. It is stated that the amount the employee contributes to the fund is 9% of $45,624. $45,624(0.09) = $4,106.16 The employer contributes a maximum of 3% of the employee contribution. Therefore: $45,624(0.03) = $1368.72. Therefore, the total annual contribution is $5,474.88, giving a future value of $196,302.40. For the Roth IRA, the monthly contributions are $352.45 giving a future value of $152,636.09. Therefore, the 401(k) has a greater future value by $43,666.31.
The 401(k) has a greater future value by $43,666.31.
What is future value?The worth of a current asset at some point in the future based on an estimated rate of growth is known as future value (FV).
To determine the future value for the 401(k), we need to determine the amount contributed annually.
It is stated that the amount the employee contributes to the fund is 9% of $45,624. $45,624(0.09) = $4,106.16
The employer contributes a maximum of 3% of the employee contribution. Therefore:
$45,624(0.03) = $1368.72.
Therefore, the total annual contribution is $5,474.88, giving a future value of $196,302.40.
For the Roth IRA,
the monthly contributions are $352.45 giving a future value of $152,636.09.
Therefore, the 401(k) has a greater future value by $43,666.31.
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How do you know when a situation involves a percent increase or a
percent decrease?
Answer:
1. Subtract starting value minus final value.
2.divide that amount by the absolute value of the starting value.
3. Multiplt by 100 to get a percent decrease.
4. if the percentage is negative, it means there was an increase and not an decrease
Answer:
Step-by-step explanation:
A percent of increase (or decrease) is a ratio of the amount of increase (or decrease) to the original amount. To find the percent of increase (decrease): Determine the amount of increase (or decrease). Divide this result by the given original amount.
Felipe split 7/8 pounds of candy among 6 people.
What is the unit rate in pounds per person?
Write your answer in simplest form.
Final answer:
To calculate the unit rate of candy per person, divide 7/8 pounds by 6 people, which results in 7/48 pounds per person. This is the unit rate in its simplest form.
Explanation:
To find the unit rate in pounds per person, we divide the total amount of candy, which is 7/8 pounds, by the number of people, which is 6.
Step 1: Determine the total weight of the candy: 7/8 pounds.
Step 2: Determine the number of people: 6 people.
Step 3: Divide the total weight by the number of people to get the unit rate:
(7/8) pounds ÷ 6 = (7/8) ÷ (6/1) = 7/8 ÷ 6/1 = 7/8 ÷ 6 = 7/48 pounds per person.
Step 4: Simplify the fraction, if possible. In this case, the fraction 7/48 is already in simplest form.
The unit rate of candy per person is 7/48 pounds.
Final answer:
7/48 pounds per person is the simplest form for the unit rate of candy per person.
Explanation:
The student is asking how to determine the unit rate of candy distributed per person if 7/8 pounds of candy were split among 6 people.
Calculation of Unit Rate:
To find the unit rate, divide the total amount of candy by the number of people:
7/8 pounds of candy ÷ 6 people = 7/48 pounds per person.This is already the simplest form, so the unit rate is 7/48 pounds per person.
Which value of x is a solution to this equation
5x^2-36x+36=0
Answer:
Exact Form:
x = 6 5 , 6
Decimal Form:
x = 1.2 , 6
Mixed Number Form:
x = 1 1 /5 , 6
Step-by-step explanation:
Move all terms to the left side and set equal to zero. Then set each factor equal to zero.
The value of x to this equation are as follows:
x = 6 or 6 /5
5x² - 36x + 36 = 0
Quadratic formula:-b ± √b² - 4ac / 2a
Using quadratic formula,
where,
a = 5
b = -36
c = 36
Therefore,
x = 36 ± √(-36)² - 4 × 5 × 36 / 2 × 5
x = 36 ± √1296 - 720 / 10
x = 36 ± √576 / 10
x = 36 ± 24 / 10
x = 36 + 24 / 10 or 36 - 24 / 10
x = 60 / 10 or 12 / 10
x = 6 or 6 / 5
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Which graph has a rate of change equal to
1 in the interval between 0 and 3 on the x-axis?
bo
Answer: -2
Step-by-step explanation:
Because I know
Use formulas to find the lateral area and surface area of the given prism. Round
your answer to the nearest whole number.
4 m
13 m
Not drawn to scale
208 m²: 240 m2
136 m²: 240 m2
136 m²: 188 m²
208 m²: 188m²
The lateral area of the rectangular prism is calculated to be 208m² using the formula for lateral area of a prism. Assuming the width is the same as the length, the surface area is calculated to be 240m² using the formula for surface area of rectangular prism.
Explanation:In mathematics, the lateral area and surface area of a rectangular prism can calculated using specific formulas. To calculate the lateral area of a rectangular prism, which is the area excluding the base and the top, you use the formula: Lateral area = Perimeter of the base x Height. Given the base length is 4m, and the height is 13m, the perimeter of the base is 4 x 4m = 16m. So, the lateral area = 16m x 13m = 208m².
The surface area of a rectangular prism is calculated using the formula: Surface area = 2lw + 2lh + 2wh. where l is the length, w is the width, and h is the height. In this case, the width is not given. Assuming it's the same as the length (since it's also 4m in the given figure), the surface area = 2*(4m*4m) + 2*(4m*13m) + 2*(4m*13m) = 32m² + 104m² + 104m² = 240m².
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Ms. Bedminster surveyed 6th graders at CMSP327 and asked them how many hours do they spend on certain activities each day. Mya said she spends 25% of her day at school. How many hours does she spend at school?
Answer:
6 hours
Step-by-step explanation:
Please refer to the attached image for explanations
Find the product: -4/7 (- 13/2)
Answer:
the answer is 3.71
Step-by-step explanation:
Answer:
3.71428571429
Step-by-step explanation:
PL
(05.02)
Find the measure of angle x in the figure below: (5 points)
Answer:
the measure of angle x is 80
Step-by-step explanation:
Alexis wants to make a paperweight at pottery class. He designs a pyramid-like model with a base area of 100square centimeters and a height of 6 centimeters. He wants the paperweight to weigh at least 300 grams. What is the lowest possible density of the material Alexis uses to make the paperweight?
Answer:
1.5 gm per cubic cm
Step-by-step explanation:
Volume of pyramid is given by formula = 1/3(area of base * height)
Given in problem
base area = 100 sq. cm
height= 6cm
therefore volume of the pyramid = 1/3(100*6) = 200 cubic cm
Formula of density is = mass of the body/ volume of the body
minimum weight of required paperweight = 300 gm
minimum density of paperweight = 300/200 = 1.5 gm per cubic cm
Solve for x: x - 4 = -10
Answer:
x = -6
Step-by-step explanation:
x - 4 = -10
x = -6
Can I get brainliest
Answer:
x = -6
Step-by-step explanation:
you just add 4 to the other side in order to isolate the x
check
-6 - 4 = -10
An Olympic runner set a record for the 100m dash.
The time was ten and sixty-two hundredths
seconds. How would you write this as a number?
To convert the word form of the time to decimal form, add 10.00 and 0.62 to get the final number representation of 10.62 seconds.
Explanation:To write the time of ten and sixty-two hundredths seconds as a number, we need to convert the word form into decimal form. The word form tells us that there are ten whole seconds and sixty-two hundredths of a second. To convert this to decimal form, we write 10 as 10.00 and 62 hundredths as 0.62. Then, we add the two decimal numbers together to get the final result: 10.00 + 0.62 = 10.62. Therefore, the number representation of the given time is 10.62 seconds.
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What is the lateral area of the rectangular prism? Round the answer to the nearest whole number. Assume the 4 m × 8 m rectangles on the left and right are the bases. The diagram is not drawn to scale.
Answer:
432 square meters
Step-by-step explanation:
The lateral area means the area that's not the bases. We have to add up all the lateral faces.
2(big rectangle area) + 2(small rectangle area) = total lateral area
Big rectangle area: A = lw = 8*18 = 144 square metersSmall rectangle area: A = lw = 18*4 = 72 square meters2(144) + 2(72) = total lateral area
total lateral area = 432 square meters
Answer:
432
Step-by-step explanation:
432
An object is thrown up word from the top of 160 foot building with an initial velocity of 144 ft./s. The height H of the object after T seconds is given by the quadratic equation h= -16t^2 + 144t + 160. When will the object hit the ground?
Answer:
After 10 seconds
Step-by-step explanation:
In this problem, the height of the object after t seconds is described by the function
[tex]h(t)=-16t^2+144t+160[/tex]
where
160 ft is the initial height of the object at t = 0
+144 ft/s is the initial velocity
[tex]-32 ft/s^2[/tex] is the acceleration due to gravity (downward)
Here we want to find the time at which the object hits the ground, so the time t at which
[tex]h(t)=0[/tex]
Therefore we can write
[tex]-16t^2+144t+160 =0[/tex]
Simplifying (dividing each term by 16), we get
[tex]-t^2+9t+10=0[/tex]
This is a second-order equation in the form
[tex]ax^2+bx+c=0[/tex]
Which has solutions given by the formula
[tex]t_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex] (2)
Here we have:
a = -1
b = 9
c = 10
Substituting into (2) we find the solutions:
[tex]t_{1,2}=\frac{-9\pm \sqrt{9^2-4(-1)(10)}}{2(-1)}=\frac{-9 \pm 11}{-2}[/tex]
Which gives:
[tex]t_1=10 s\\t_2 =-1 s[/tex]
Since time cannot be negative, the only solution is
t = 10 seconds
Write the expression in standard form 6(-5r-4)-2(r-7s-3)
Final answer:
To write the expression 6(-5r-4)-2(r-7s-3) in standard form, distribute the numbers outside the parentheses across the terms within and combine like terms to get -32r + 14s - 18.
Explanation:
To write the expression 6(-5r-4)-2(r-7s-3) in standard form, we must distribute the numbers outside the parentheses to each term inside the parentheses and then combine like terms.
Distribute the 6: 6 × (-5r) = -30r and 6 × (-4) = -24, giving us -30r - 24.
Distribute the -2: -2 × r = -2r, -2 × (-7s) = +14s and -2 × (-3) = +6, giving us -2r + 14s + 6.
Combine the two resulting expressions: -30r - 24 - 2r + 14s + 6.
Combine like terms: The terms -30r and -2r combine to -32r, and the constants -24 and +6 combine to -18.
The standard form of the expression is -32r + 14s - 18.
Check the answer to see if it is reasonable.
A recipe for beef stew calls for 1 pound of beef and 3 potatoes. The recipe is doubled to include 2 pounds of beef and 6 potatoes. Given that the proportion 1 3 = 2 6 represents the situation, which of these statements are true? Check all that apply.
Answer:
b and d
Step-by-step explanation:
1. The cities of Dallas, Charleston, and Indianapolis form a triangle. The distance from Dallas to
Charleston is 980 mies, Charleston to Indianapolis is 595 mies, and Indianapolis to Dallas is 764
mies. If Dallas and Charleston lie on the same latitude, find the bearing from Charleston to
Indianapolis.
Answer: 130 degree
Step-by-step explanation:
Given that
The distance from
Dallas to Charleston = 980 miles Charleston to Indianapolis = 595 miles,
Indianapolis to Dallas is 764 mies.
If Dallas and Charleston lie on the same latitude, find the bearing from Charleston to Indianapolis.
Let us use cosine formula to find the angle at Charleston
764^2 = 980^2+595^2-2(980)(595)cosØ
583696 = 1314425-1166200cosØ
-1166200cosØ = -730729
CosØ = 0.6266
Ø = 51.2
Let us also calculate the angle at Indianapolis by using sine rule
980/sinI = 764/sin 51.2
Reciprocal both sides
SinI/980 = sin 51.2/764
Sin I = 980 × sin52.1/764
Sin I = 0.9996
I = 88.5 degree
The bearing = 270 - ( 51.2 + 88.5)
= 270 - 140
= 130 degree.
In this exercise we have to use the knowledge of triangles to calculate the angle of this geometric figure, in this way we can say that:
130 degree
Given that the distance from dallas to Charleston and Charleston to Indianapolis and Indianapolis to Dallas, we can use the the cosine formula to find the angle at Charleston, so we have:
[tex]764^2 = 980^2+595^2-2(980)(595)cos\phi\\583696 = 1314425-1166200cos\phi\\-1166200cos\phi = -730729\\Cos\phi = 0.6266\\\phi = 51.2[/tex]
Let us also calculate the angle at Indianapolis by using sine rule:
[tex]980/sin\theta = 764/sin (51.2)\\Sin\theta/980 = sin 51.2/764\\Sin\theta = 980 * sin52.1/764\\Sin \theta = 0.9996\\\theta = 88.5 degree = 270 - ( 51.2 + 88.5)\\= 270 - 140\\= 130 degree.[/tex]
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ellens dog has a pen that is 3
Answer:
36 ft^2
Step-by-step explanation:
3^2*3^3=9+27=36
Answer:
36 ft^2
Step-by-step explanation:
What is the measure of each exterior angle of a regular 30-gon?
Answer:
Each Exterior Angle = (360 degrees) ÷ (Number of Sides)
Each Exterior Angle = 360 / 30
Each Exterior Angle = 12 degrees
Step-by-step explanation:
The measure of each exterior angle of a regular 30-gon will be 12°.
Based on the information, it should be noted that the value of the total angles will be 360°.
Therefore, the measure of each exterior angle of a regular 30-gon will be:
= 360/30
= 12°
Therefore, the measure of each exterior angle of a regular 30-gon is 12°.
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what is an example of constant terms
A constant term is the whole number in an expression!
They are always easy to identify
5x + 14 ....
14 is the constant term!
80y - 2
-2 is the constant term!
I hope I helped :)
[tex]\huge\boxed{\sf{Hello\;there! :)}}[/tex]
A constant term is a number in an algebraic expression or equation.
A constant doesn't have any variables next to it.
For example, in the equation
4x²-2x=24, 24 is the constant.
[tex]\huge\boxed{\text{Hope\;it\;helps\;you!}}[/tex]
[tex]\huge\bold{\underline{\underline{Good\;luck!}}}[/tex]
[tex]\huge\rm{StillEnjoyingLife :)[/tex]
Clare made $160 babysitting last summer. She puts the money in the saving account that pays 3% interest per year. If Clare doesn't touch the money in her account, she can find the amount she'll have the next year by multiplying her current amount by 1.03. How much money will Clare have in her account after 1 year?
Answer:
Clare account balance after one year will be $164.80
Step-by-step explanation:
Interest = PRT/100
Principal P = $160, Rate r = 3%, and time t = 1 year
Interest = (160*3*1)/100
Interest =480/100
= $4.80
The total balance in Clare's account after 1 year will be $160 + $4.80
= $164.80
Explain the steps to take when writing a coordinate geometry proof.
Final answer:
To write a coordinate geometry proof, first choose and plot your figures in a coordinate system, then use properties of the system to express geometric properties, apply Euclid's geometric axioms, perform algebraic manipulations, and compile these into a logical proof sequence.
Explanation:
Writing a coordinate geometry proof involves several logical steps that combine geometry with algebra. The aim is to use the properties and axioms of geometry, expressed in a coordinate system, to prove a geometric statement. Here's a simplified process:
Choose a coordinate system and plot the given geometric figures using the appropriate coordinates.
Use the properties of the chosen coordinate system to write expressions for geometric properties such as the lengths of sides, slopes of lines, and coordinates of midpoints or centroids.
Apply geometric axioms, such as those from Euclid's geometry, to relate different parts of the figure.
Carry out algebraic manipulations to show that a certain property holds, such as proving two sides are congruent or two angles are equal.
Compile the findings into a logical sequence to form the proof, stating clearly how the coordinate geometry demonstrates the property or theorem in question.
Throughout your proof, ensure you reference the correct coordinate systems and datum points, and be prepared to convert between coordinate systems if necessary, which can be crucial when applying different geometric axioms or properties.
A professor has 24 pairs of socks in his drawer. 18 pairs are black and the rest are brown. He always wears black pants. He does not pay much attention to which socks he puts on each day. Supposed he selects one pair of socks on Monday and one on Tuesday. Since he has lots of clean socks in the drawer he does not do laundry. What is the likelihood that the professor's socks matched his pants on both days (i.e. he picked black socks)
Answer:
The likelihood that the professor's socks matched his pants on both days is 0.5625
Step-by-step explanation:
Total no. of pair of socks = 24
No. of black pairs = 18
We are given that he selects one pair of socks on Monday and one on Tuesday. Since he has lots of clean socks in the drawer he does not do laundry.
So, It is a case of replacement
He always wears black pants.
We are supposed to find he likelihood that the professor's socks matched his pants on both days i.e. Black socks both days.
So ,Probability of wearing black socks on Monday = [tex]\frac{18}{24}[/tex]
So ,Probability of wearing black socks on Tuesday =[tex]\frac{18}{24}[/tex]
So,the likelihood that the professor's socks matched his pants on both days = [tex]\frac{18}{24} \times \frac{18}{24}=0.5625[/tex]
Hence the likelihood that the professor's socks matched his pants on both days is 0.5625
Final answer:
The probability that the professor picks black socks on both Monday and Tuesday is 9/16.
Explanation:
The question asks about the probability that a professor will pick black socks on two consecutive days without doing laundry in between. With 24 pairs of socks, of which 18 pairs are black and 6 pairs are brown, we will calculate the likelihood of the professor picking black socks on both Monday and Tuesday.
First, the probability of picking black socks on Monday (first pick) is 18 black pairs divided by 24 total pairs, which simplifies to:
Probability of black socks on the first pick = 18/24 = 3/4Since the socks are not washed between Monday and Tuesday, the number of socks available remains the same. Therefore, the probability of picking black socks again on Tuesday (second pick) is also 3/4.
To find the overall probability that the professor picks black socks on both days, we must multiply the probability of the first event by the probability of the second event. In this case:
Overall probability = Probability on Monday × Probability on TuesdayOverall probability = 3/4 × 3/4 = 9/16The probability that the professor's socks matched his pants (picked black socks) on both days is 9/16.
Find the volume of a cone with base area 36π ft2 and a height equal to twice the radius. Give your answers both in terms of π and rounded to the nearest tenth.
The volume in terms of π is _____ft3.
The volume rounded to the nearest tenth is _____ft3.
Answer:
The volume in terms of π is 144 ft³
The volume rounded to the nearest tenth is 452.4 ft³
Step-by-step explanation:
The formula of the volume of a cone is V = [tex]\frac{1}{3}[/tex] π r² h , where r is its radius and h is its height
The formula of the area of a circle is A = π r²
∵ The area of the base of the cone is 36π ft²
- Equate the formula of the area of the circle by 36π to find r
∵ πr² = 36π
- Divide both sides by π
∴ r² = 36
- Take √ to both sides
∴ r = 6 ft
∵ The height of the cone is equal to twice the radius
∵ The radius of the cone is 6 ft
- Multiply 6 by 2 to find the height
∴ h = 2(6) = 12 ft
∴ The height of the cone is 12 ft
Substitute the values of h and r in the formula of the volume of the cone to find it
∵ V = [tex]\frac{1}{3}[/tex] π(6)²(12)
∴ V = 144 π ft³
∴ V = 452.3893421 ft³
- Round it to the nearest tenth
∴ V = 452.4 ft³
A gas pump fills 2^-3 gallon of gasoline per second. How many gallons does the pump fill in one minute?
Answer:
7.5 gallons
Step-by-step explanation:
2^-3= 0.125
0.125*60= 7.5
Final answer:
The gas pump fills 7.5 gallons of gasoline in one minute.
Explanation:
If a gas pump fills 2-3 gallon of gasoline per second, we first need to understand how much that is in gallons. 2-3 is the same as 1/23 or 1/8. Therefore, the pump fills 1/8 gallon per second.
To find out how many gallons the pump fills in one minute, we multiply the amount filled per second by the number of seconds in a minute:
Gallons per minute = gallons per second × seconds per minute
Gallons per minute = 1/8 gallon/second × 60 seconds/minute
Gallons per minute = 60/8 gallons
By reducing this fraction, we get:
Gallons per minute = 7.5 gallons
Thus, the gas pump fills 7.5 gallons of gasoline in one minute.
A particular cell phone company offers 4 models of phones, each in 6 different colors and each available with an one of 5 calling plans. How many combinations are possible?
There are 120 possible combinations of phone models, colors, and calling plans.
Explanation:To calculate the number of possible combinations of phone models, colors, and calling plans, we need to multiply the number of choices for each category together. In this case, there are 4 phone models, 6 colors, and 5 calling plans. So the total number of combinations is 4 x 6 x 5 = 120.
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It takes a smaller hose twice as long to fill a certain swimming pool as it does a larger hose. It takes both hoses working together 25 minutes to fill the pool. How long will it take the larger hose to fill the pool by itself?
Answer: it will take the larger hose 37.5 minutes to fill the pool.
Step-by-step explanation:
Let t represent the number of minutes it will take the larger hose to fill the pool by itself. It means that the rate at which larger hose fills the pool per minute is 1/t
It takes the smaller hose twice as long to fill the swimming pool as it does the larger hose. It means that the number of hours it takes smaller hose to fill the pool is 2t and it fills it alone, then the rate at which it fills the pool per minute is 1/2t
By working together, they would work simultaneously and their individual rates are additive. It takes both hoses working together 25 minutes to fill the pool. It means that the rate at which they fill the pool together per minute is 1/25. Therefore,
1/t + 1/2t = 1/25
3/2t = 1/25
Cross multiplying, it becomes
2t = 3 × 25 = 75
t = 75/2 = 37.5 minutes
Final answer:
To find the time it takes for the larger hose to fill the pool alone, we set up an equation based on their combined work rate and solve for the larger hose's time, finding that it would take 75 minutes.
Explanation:
The question is about determining how long it will take for the larger hose alone to fill the pool, given that it takes twice as long for a smaller hose to complete the same task, and together they can fill the pool in 25 minutes. To solve this problem, let's denote the time it takes for the larger hose to fill the pool by itself as x minutes. Therefore, the smaller hose takes 2x minutes to fill the pool by itself.
Since the problem involves rates of work, we can use the formula Work done = Rate × Time. The rate can be considered as the portion of the pool filled per minute. So, the larger hose's rate is 1/x and the smaller hose's rate is 1/(2x). Working together, their combined rate is 1/25 of the pool per minute, following the equation 1/x + 1/(2x) = 1/25.
By solving this equation: 2/2x + 1/2x = 1/25 ⇒ 3/(2x) = 1/25 ⇒ x = 75. Therefore, it would take the larger hose 75 minutes to fill the pool by itself.