Answers
㏒₆(17) + ㏒₆(w)
It is a sum.
[tex]log_6(17w)[/tex] can be expressed as [tex]log_6(17) + log_6(w)[/tex] using the sum of logarithms property.
To express log base 6 of 17w as the sum of logarithms, you can use the properties of logarithms, specifically the product rule for logarithms:
[tex]log_b(xy) = log_b(x) + log_b(y)[/tex]
In this case, you have log base 6 of 17w, and you want to express it as the sum of logarithms. You can do this as follows:
[tex]log_6(17w) = log_6(17) + log_6(w)[/tex]
So, log base 6 of 17w can be expressed as the sum of log base 6 of 17 and log base 6 of w.
This is a straightforward application of the product rule for logarithms, where you split the logarithm of a product into the sum of the logarithms of the individual factors.
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Related Questions
Suppose that the mean GRE score for the USA is 500 and the standard deviation is 75. Use the Empirical Rule (also called the 68-95-99.7 Rule) to determine the percentage of students likely to get a score between 350 and 650? What percentage of students will get a score above 500?
Answers
Find how many std deviations 350 and 650 are from the mean of 500.
350 - 500 = -150 ----> -150/75 = -2
650 - 500 = 150 ----> 150/75 = 2
Each are 2 std deviations from the mean. This means that about 95% of students are between 350 and 650.
By definition, 50% will be above the mean and below the mean.
Therefore 50% of the students will get a score above 500.
Using the Empirical Rule, it is determined that 95% of students are likely to get a GRE score between 350 and 650, and 50% of students will get a score above 500.
According to the Empirical Rule (also known as the 68-95-99.7 Rule), about 95% of the values in a data set with a normal distribution will fall within two standard deviations of the mean. In this case, the mean GRE score is 500 and the standard deviation is 75. Using this rule, we can determine that 95% of the students would likely score between 500 - (2 imes 75) and 500 + (2 imes 75), equating to scores between 350 and 650.
Regarding the percentage of students that will get a score above 500, the distribution is symmetric around the mean in a normal distribution. Since the mean is 500, 50% of the values will lie above the mean and 50% will lie below. Therefore, 50% of students are expected to score above 500.
Make a down payment of $1500 and finance the rest of $20000 at 1.9% interest rate, making equal monthly payments for 5 years.
Calculate the total interest you will pay over the term of the loan. Show the computations that lead to your answer.
Answers
1 year = 12 months
12 * 5 = 60 months
Therefore, in 5 years there are 60 months.
Now lets solve this problem step by step.
Subtract the down payment from $20,000
$20000-$1500=$18500
Multiply the remaining number by the interest rate.
$18500 *1.9 = $35150
Divide 35150 by number of months in 5 years (60)
$35150 / 60 = $585
Therefore, you have to pay $585 per month.
The sum of the length of l and 15
Answers
Write the given expression in terms of x and y only.
sin(sin−1(x) + cos−1(y))
Answers
_______________
Write the expression below in terms of x and y only:
(I'm going to call it "E")
[tex]\mathsf{E=sin\!\left[sin^{-1}(x)+cos^{-1}(y)\right]\qquad\quad(i)}[/tex]
Let
[tex]\begin{array}{lcl} \mathsf{\alpha=sin^{-1}(x)}&\qquad&\mathsf{then~-\,\dfrac{\pi}{2}\le \alpha\le \dfrac{\pi}{2}}\\\\\\ \mathsf{\beta=cos^{-1}(x)}&\qquad&\mathsf{then~0\le \beta\le \pi.} \end{array}[/tex]
so the expression becomes
[tex]\mathsf{E=sin(\alpha+\beta)}\\\\ \mathsf{E=sin\,\alpha\,cos\,\beta+sin\,\beta\,cos\,\alpha\qquad\quad(ii)}[/tex]
• Finding [tex]\mathsf{sin\,\alpha:}[/tex]
[tex]\mathsf{sin\,\alpha=sin\!\left[sin^{-1}(x)\right]}\\\\ \mathsf{sin\,\alpha=x\qquad\quad\checkmark}[/tex]
• Finding [tex]\mathsf{cos\,\alpha:}[/tex]
[tex]\mathsf{sin^2\,\alpha=x^2}\\\\ \mathsf{1-cos^2\,\alpha=x^2}\\\\ \mathsf{cos^2\,\alpha=1-x^2}\\\\ \mathsf{cos\,\alpha=\sqrt{1-x^2}\qquad\quad\checkmark}[/tex]
because [tex]\mathsf{cos\,\alpha}[/tex] is positive for [tex]\mathsf{\alpha\in \left[-\frac{\pi}{2},\,\frac{\pi}{2}\right].}[/tex]
• Finding [tex]\mathsf{cos\,\beta:}[/tex]
[tex]\mathsf{cos\,\beta=cos\!\left[cos^{-1}(y)\right]}\\\\ \mathsf{cos\,\beta=y\qquad\quad\checkmark}[/tex]
• Finding [tex]\mathsf{sin\,\beta:}[/tex]
[tex]\mathsf{cos^2\,\alpha=y^2}\\\\ \mathsf{1-sin^2\,\beta=y^2}\\\\ \mathsf{sin^2\,\beta=1-y^2}\\\\ \mathsf{sin\,\beta=\sqrt{1-y^2}\qquad\quad\checkmark}[/tex]
because [tex]\mathsf{sin\,\beta}[/tex] is positive for [tex]\mathsf{\beta\in [0,\,\pi].}[/tex]
Finally, you get
[tex]\mathsf{E=x\cdot y +\sqrt{1-y^2}\cdot \sqrt{1-x^2}}\\\\\\ \therefore~~\mathsf{sin\!\left[sin^{-1}(x)+cos^{-1}(y)\right]=x\cdot y +\sqrt{1-y^2}\cdot \sqrt{1-x^2}\qquad\quad\checkmark}[/tex]
I hope this helps. =)
Tags: inverse trigonometric trig function sine cosine sin cos arcsin arccos sum angles trigonometry
A restaurant offers two different kinds of soup and three different kinds of salad.
(a) If you are having either soup or salad, how many choices do you have?
(b) If you are having both soup and salad, how many choices do you have?
Answers
b) There could be 6 different choices
a. There could be 5 different choices and b. There could be 6 different choices.
What is the fundamental principle of multiplication?Multiplication is the mathematical operation that is used to determine the product of two or more numbers. If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
Given tha the number of different kinds of soups = 2
The number of different kinds of salad = 3
(a) If you are having either soup or salad, then the total number of choices you have;
2 + 3 = 5
Hence, the total number of choices you have 5.
(b) If you are having both soup and salad, then the number of choices you have;
2 x 3 = 6
Hence, the total number of choices you have 6.
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How to change 2/3 to eighteenths
Answers
What is the GCF of 3 and 18
3: 1,3
18: 1,2,3,6,9,18
GCF-3
Therefore, 2/3 = 12/18 because 3 × 6= 18
what you do to the denominator you have to do the numerator
so 2 × 6= 12
The eighteenths of 2/3 is 12/18.
What is multiplication?In mathematics, multiplication is a method of finding the product of two or more numbers. It is one of the basic arithmetic operations, that we use in everyday life.
here, we have,
let,
2/3 = n/18
What is the GCF of 3 and 18
3: 1,3
18: 1,2,3,6,9,18
GCF-3
Therefore, 2/3 = 12/18
because 3 × 6= 18
what you do to the denominator you have to do the numerator
so 2 × 6= 12.
hence, eighteenths of 2/3 is 12/18.
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A building in a city has a rectangular base. The length of the base measures 70ft less than twice the width. The perimeter of the base is 910 ft. What are the dimensions of the base?
Answers
width=x
perimeter of a rectangle=2(base)+2(width)
We can suggest the next equation:
2(2x-70)+2x=910
4x-140+2x=910
6x=910+140
6x=1050
x=1050/6
x=175
base: 2x-70=2(175)-70=350-70=280
Answer,: the length of the base would be 280 ft.
two angles are supplementary. their difference is 22°. find the angles
Answers
x+y = 180 (supplementary)
x - y = 22(given) so x = 22 + y
22+ y + y = 180
2y + 22 = 180
2y =180 -22
2y = 158
y = 158/2
y = 79
x = 22 + y = 22 + 79 = 101
answer:Two angles are supplementary angles are 101 and 79
Write a number from 20 to 50 that has both tens and ones
Answers
Answer:
45
Step-by-step explanation:
We know that,
If we have the numbers of the form xyz.
This gives that, x is placed at the hundredth place, y is placed at the tens place and z is located at ones place.
Since, we need a number between 20 and 50 having both tens and ones place.
We can take any number from 20 to 50.
For e.g. Let us take 45.
In 45, 4 is placed at the tens place and 5 is placed at the ones place.
Hence, 45 is the number between 20 to 50 having both tens and ones place.
A lock has a disk with 36 numbers written around its edge. The combination to the lock is made up of three numbers. What is the probability of correctly guess the combination?
Answers
The probability of correctly guessing the combination will be 1/7140.
What are permutation and combination?A permutation is an act of arranging items or elements in the correct order. Combinations are a way of selecting items or pieces from a group of objects or sets when the order of the components is immaterial.
A lock has a disk with 36 numbers written around its edge. The combination of the lock is made up of three numbers.
The total number of the events will be
Total event = ³⁶C₃
Total event = 36! / [(36 - 3)! x 3!]
Total event = 7140
The favorable number of the event will be one.
Then the probability of correctly guessing the combination will be
P = 1 / 7140
The probability of correctly guessing the combination will be 1/7140.
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Find the exact solution using common logarithms 2^5x+3= 3^2x+1
Answers
[tex]log_{2}(2^{5x + 3}) = log_{2}(3^{2x + 1})[/tex]
[tex](5x + 3)log_{2}(2) = (2x + 1)log_{2}(3)[/tex]
[tex]5x + 3 = 1.58(2x + 1)[/tex]
[tex]5x + 3 = 3.16x + 1.58[/tex]
[tex]1.84x + 3 = 1.58[/tex]
[tex]1.84x = -1.42[/tex]
[tex]x \approx -0.77[/tex]
Find the value of I given that p=300,r=0.14,andt=2
Answers
P = 300
R = 0.14
T = 2
now it is just a matter of subbing and doing the math
I = (300)(0.14)(2)
I = 84 <==
the kilo-prefix in the metric system means __________________ units
Answers
A department store holds a year-end clearance sale that includes a 14.5% discount on cosmetics. Find the sales price of the bottle of perfume if it's original price was 56.69
Answers
56.69 - 0.145(56.69)
= 56.69 - 8.22
= 48.47
How to calculate discount and sale price?Find the original price (for example $90 )Get the discount percentage (for example 20% )Calculate the savings: 20% of $90 = $18.Subtract the savings from the original price to get the sale price: $90 - $18 = $72.the formula for discount price
Subtract the final price from the original price. Divide this number by the original price. Finally, multiply the result by 100. You've obtained a discount in percentages.
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your classmate says he can write the equation of a quadratic function that passes through the points (3,4), (5,-2),and (3,0). Explain his error.
Answers
In this case, (3,4) and (3,0). There can't be two exact same value in a slope.
The classmate's claim is incorrect, as a parabola cannot pass through two points with the same x-coordinate but different y-values, making it impossible for a single quadratic function to pass through points (3,4), (5,-2), and (3,0).
The error in your classmate's assertion that he can write the equation of a quadratic function that passes through the points (3,4), (5,-2), and (3,0) lies in the fact that a quadratic function is a parabola, and no parabola can pass through two points that have the same x-coordinate but different y-values. In other words, any vertical line (like x=3) can intersect a parabola at most once because a quadratic function has only one y value for each x value. Since (3,4) and (3,0) both have the x-coordinate of 3 but different y-coordinates, they cannot both lie on the same parabola. The equation of a quadratic function is typically written as [tex]y = ax^2 + bx + c[/tex], where a, b, and c are constants. To pass through specific points, these constants must be determined such that the function satisfies the coordinates of the points. However, due to the reason explained, no set of constants will satisfy the condition for the mentioned points.
alana bought 6 books each book costs 13$ and each bookmark cost 2$ how much did she spend on books
Answers
2 times 6 is 12
12 plus 78 is 90
In our solar system, 666 of the 888 planets have moons. What percentage of the planets have moons?
Answers
Answer:
75% of the planets have moons
Step-by-step explanation:
In order to calculate this you just have to do a simple rule of three, in which you will use 888 as the 100 percent and you want to discover what percentage of that does 666 represents:
[tex]\frac{888}{100} \frac{666}{x} \\x=\frac{666*100}{888}\\ x=75\\[/tex]
So you have that 666 is the 75% of 888, that means that 75% of the planets have moons.
a bag contains five batteries, all the same size and equally likely to be selected. Each battery is a different brand. If you select three batteries at random, use the counting principle to determine how many points will be in the sample space if the batteries are selected with replacement and without replacement?
Answers
When selecting three batteries at random with replacement, there are 125 points in the sample space. When selecting three batteries at random without replacement, there are 60 points in the sample space.
Explanation:To determine the number of points in the sample space when selecting three batteries at random with replacement, we use the counting principle. Since there are 5 batteries, each with an equal likelihood of being selected, and we are selecting three batteries, we can calculate the number of points in the sample space as follows:
Number of points = Number of possible outcomes for the first battery * Number of possible outcomes for the second battery * Number of possible outcomes for the third battery
Since there are 5 batteries, the number of possible outcomes for each battery is 5. Therefore,
Number of points = 5 * 5 * 5 = 125
So, there are 125 points in the sample space when selecting three batteries at random with replacement.
When selecting three batteries at random without replacement, the number of points in the sample space will be different. We can calculate it using a similar approach, but taking into account that once a battery is selected, it is removed from the bag and cannot be selected again. Therefore, for the first battery, there are 5 possible outcomes. For the second battery, there will be 4 possible outcomes since one battery has already been selected. For the third battery, there will be 3 possible outcomes. Therefore,
Number of points = 5 * 4 * 3 = 60
So, there are 60 points in the sample space when selecting three batteries at random without replacement.
The number of possible outcomes for selecting three batteries with replacement is 125. For selecting three batteries without replacement, the total is 60. The counting principle multiplies the number of options for each choice to calculate the sample space.
Explanation:When selecting three batteries from a bag containing five different brands at random, the number of points in the sample space can be determined using the counting principle. The sample space represents all possible outcomes of an experiment.
a. Sampling with replacement
When batteries are selected with replacement, after each selection, the chosen battery is put back into the bag, making it available for the next selection. Thus, for each pick of the three batteries, there are five choices. The total number of points in the sample space for selecting three batteries with replacement is therefore 5 × 5 × 5 = 125.
b. Sampling without replacement
When batteries are selected without replacement, the chosen battery is not put back in the bag, which means the options decrease with each selection. The first pick has five choices, the second pick will have four (since one battery is already picked), and the third pick will have three. The total number of points in the sample space for selecting three batteries without replacement is therefore 5 × 4 × 3 = 60.
Crafty grandma Edith set her family down during Thanksgiving and told them they couldn't have any pet can pass until they worked out this puzzle are six-year-old granddaughter was the first to solve it can you work out what 9183 equals
8809=6 3590=2
7111=0 6855=3
9881=5 1012=1
6660=4 5731=0
5531=0 9191=2
2516=1 9193=?
Answers
how many different ways can you make 82 cents using current u.s. currency
Answers
You need non-negative integer solutions to p+5n+10d+25q = 82.
If p = leftovers, then you simply need 5n + 10d + 25q ≤ 80.
So this is the same as n + 2d + 5q ≤ 16
So now you simply have to "crank out" the cases.
Case q=0 [ n + 2d ≤ 16 ]
Case (q=0,d=0) → n = 0 through 16 [17 possibilities]
Case (q=0,d=1) → n = 0 through 14 [15 possibilities]
...
Case (q=0,d=7) → n = 0 through 2 [3 possibilities]
Case (q=0,d=8) → n = 0 [1 possibility]
Total from q=0 case: 1 + 3 + ... + 15 + 17 = 81
Case q=1 [ n + 2d ≤ 11 ]
Case (q=1,d=0) → n = 0 through 11 [12]
Case (q=1,d=1) → n = 0 through 9 [10]
...
Case (q=1,d=5) → n = 0 through 1 [2]
Total from q=1 case: 2 + 4 + ... + 10 + 12 = 42
Case q=2 [ n + 2 ≤ 6 ]
Case (q=2,d=0) → n = 0 through 6 [7]
Case (q=2,d=1) → n = 0 through 4 [5]
Case (q=2,d=2) → n = 0 through 2 [3]
Case (q=2,d=3) → n = 0 [1]
Total from case q=2: 1 + 3 + 5 + 7 = 16
Case q=3 [ n + 2d ≤ 1 ]
Here d must be 0, so there is only the case:
Case (q=3,d=0) → n = 0 through 1 [2]
So the case q=3 only has 2.
Grand total: 2 + 16 + 42 + 81 = 141
Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible
Answers
Dawn took two pies to a school bake. She cut Pie A into 8 slices and Pie B into 6 slices. She sold two slices of Pie A and three slices in Pie B, how much total Pie Dawn has remaining?
Answers
Dawn has 6 slices of Pie A and 3 slices of Pie B remaining.
Explanation:Dawn brought two pies to a bake sale. Pie A was cut into 8 slices and Pie B was cut into 6 slices. She sold 2 slices of Pie A and 3 slices of Pie B. To calculate how much pie Dawn has remaining, we need to subtract the slices sold from the total slices in each pie. So, for Pie A, we have 8 slices - 2 slices = 6 slices remaining. And for Pie B, we have 6 slices - 3 slices = 3 slices remaining. Therefore, Dawn has a total of 6 slices of Pie A and 3 slices of Pie B remaining.
Final answer:
Dawn originally had two pies, with Pie A cut into 8 slices and Pie B cut into 6 slices. She sold two slices from Pie A and three from Pie B, leaving her with a total of 9 slices remaining from both pies.
Explanation:
To determine how much total Pie Dawn has remaining after selling some slices, we can use simple subtraction.
Dawn originally had two pies:
Pie A was cut into 8 slices, and she sold 2 slices.
Pie B was cut into 6 slices, and she sold 3 slices.
To find out how much of each Pie is left, we subtract the number of slices sold from the total number of slices:
Pie A remaining: 8 - 2 = 6 slices
Pie B remaining: 6 - 3 = 3 slices
To find the total amount of Pie remaining, we add the slices remaining from Pie A and Pie B:
Total Pie remaining: 6 (from Pie A) + 3 (from Pie B) = 9 slices
Dawn is left with 9 slices of Pie in total after the sales.
If the equilibrium interest rate in the money market is 5%, then at an interest rate of 2% sellers of interest-bearing financial assets _____ interest rates to find willing buyers.
Answers
If the equilibrium interest rate in the money market is 5%, then at an interest rate of 2% sellers of interest-bearing financial assets greater than interest rates to find willing buyers.
Final answer:
At an interest rate of 2%, below the equilibrium rate of 5%, sellers of interest-bearing financial assets would need to raise interest rates to find willing buyers and reach the equilibrium.
Explanation:
If the equilibrium interest rate in the money market is 5%, then at an interest rate of 2%, sellers of interest-bearing financial assets would need to raise interest rates to find willing buyers. This situation outlines how the market responds when the interest rate is below the equilibrium level. If the interest rate is below equilibrium, there is an excess demand for financial capital; too many borrowers and not enough lenders at this low rate. To attract more lenders (sellers of financial assets), the interest rate would need to be increased. Conversely, when the interest rate is above the equilibrium, there is an excess supply, meaning there are more suppliers of loans than there are borrowers. Sellers in this scenario lower interest rates to attract more borrowers, bringing the market back to equilibrium.
This concept is parallel to the example where, at an interest rate of 21%, the supplied funds increase greatly while the demand falls sharply, leading to excess supply. Suppliers of financial capital, such as credit card companies, will then reduce their rates to encourage more borrowing, pushing the interest rate back toward the equilibrium. In the case of a government borrowing increase, if it leads to a higher demand for financial capital, this would cause the demand curve to shift to the right and result in a higher interest rate, potentially crowding out private investment.
fibd the length of the strapping needed to secure the pipe in the corner of the room as shown. the distance from a to b is 5.657 feet.
Answers
(the triangle is isosceles.)
2x^2 = 32
so
x^2 = 16
x=4
which is the radius of the circle!
If the strapping is 1/4 the circumference and c= 2 pi r
then the strapping is 2( 3.1416) 4 /4
or 6.2832 inches
A 200 centimeter-long wire is cut into three pieces. The second piece is 3 times as long as the first. The third piece is 2 times as long as the second. How long is each piece?
Answers
x + y + z = 200 since the sum of the lengths equals the total length of the wire
y = 3x since piece two is 3 times longer than piece one
z = 2y = (2)(3)(x) = 6x since piece three is 2 times longer than piece two
Now we have (x,y,z) in terms of x alone, so we can substitute into the first equation:
x + 3x + 6x = 10x = 200 which means x = 20.
Then, substitute this value into all the other equations.
y = 3(20) = 60
z = 6(20) = 120
This can be checked by summing them up like this: 20 + 60 + 120 = 200.
The lengths of the three wire pieces are 20 cm for the first piece, 60 cm for the second piece, and 120 cm for the third piece.
Explanation:A wire that is 200 cm long is cut into three pieces.
We are told that the second piece is 3 times as long as the first piece and the third piece is 2 times as long as the second piece.
Let's denote the first piece's length as x centimeters.
According to the problem, the second piece will then be 3x centimeters long, and the third piece will be 2 * 3x = 6x centimeters long.
To find the length of each piece, we need to set up the following equation based on the total length of the wire:
x + 3x + 6x = 200
Combining terms, we have:
10x = 200
Dividing both sides by 10, we find:
x = 20
Therefore, the first piece is 20 cm long, the second piece is 3 * 20 cm = 60 cm long, and the third piece is 6 * 20 cm = 120 cm long.
From a marginal analysis perspective, what is the inventory carry cost for Andrews if the company carries one additional unit of Able in inventory at the end? Select: 1Save Answer $9.98 $1.20 $1.92 $3.85
Answers
whats between 0 and [tex] \pi /2[/tex]
Answers
A brace for a shelf has the shape of a right triangle. Its hypotenuse is 10 inches long and the two legs are equal in length. How long are the legs of the triangle?
Answers
45-45-90
1-1-sqrt2
Where the hypotenuse corresponds to sqrt2.
Now, your hypotenuse is 10.
To figure out what each leg is, divide 10/sqrt2 (because sqrt2/sqrt2 = 1, which is a leg length in the explanation above).
Problem: you can't divide by radicals. So, we'll have to rationalize the denominator:
(10•sqrt2)/(sqrt2•sqrt2)
This can be rewritten:
10sqrt2/sqrt(2•2)
=10sqrt2/sqrt4
=10sqrt2/2
=5sqrt2 Hope this helps!!
a printer charges a fixed setup cost plus $0.40 for every 40 copies. if 320 copies cost $43.20 how much will it cost to print 1200 copies
Answers
The cost to print 1200 copies is $52.00.
The given data tells us that 320 copies cost $43.20. Each set of 40 copies costs $0.40, so 320 copies are 8 sets (320/40). The variable cost for 320 copies is 8 × $0.40 = $3.20.
We can then determine the fixed setup cost by subtracting the variable cost from the total cost for 320 copies: $43.20 - $3.20 = $40.00.
The total cost C for printing x copies can be expressed as:
C = fixed setup cost + variable cost per set × (x/40)
So the total cost for 1200 copies is:
C = $40.00 + $0.40 × (1200/40)
Calculating this, we get:
C = $40.00 + $0.40 × 30 = $40.00 + $12.00 = $52.00
Therefore, the cost to print 1200 copies is $52.00.
At noon, ship A is 170 km west of ship B. Ship A is sailing east at 30 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM?
Answers
Final answer:
To find how fast the distance between two ships is changing, we calculate the distance they have covered, form a right triangle, and use the Pythagorean theorem and related rates to find that the distance is changing at approximately 8.94 km/h at 4:00 PM.
Explanation:
To determine how quickly the distance between ship A and ship B is changing, we can use the concept of related rates from calculus. At noon, the ships are 170 km apart. Ship A travels east (towards ship B) at 30 km/h, and ship B travels north at 25 km/h. By 4:00 PM, four hours have passed.
First, let's calculate the distance each ship has traveled after 4 hours. Ship A would have covered 30 km/h * 4h = 120 km eastward. Ship B would have traveled 25 km/h * 4h = 100 km northward.
Next, we need to calculate the distance between the two ships at 4:00 PM. Since ship A is moving east and ship B is moving north, they are traveling at right angles to each other. The distance between the two ships forms the hypotenuse of a right triangle, with the sides being the distances traveled by each ship. Using the Pythagorean theorem, the distance between the two ships at 4:00 PM (D) is:
² = (170 km - 120 km)² + (100 km)²
² = (50 km)² + (100 km)²
² = 2500 km² + 10000 km²
= √12500 km²
= 111.80 km (approx)
Finally, we have to find the rate at which this distance is changing. Let x be the east-west distance from ship B to ship A, and y be the northward distance ship B has covered. Taking the derivative of the Pythagorean relation we have, x² + y² = ², with respect to time (t) gives us:
2x(/) + 2y(/) = 2(/)
Given that dx/dt = -30 km/h (since it's decreasing as ship A sails east) and dy/dt = 25 km/h, we can solve for d/dt:
2*50*(-30) + 2*100*(25) = 2*111.80*(d/dt)
-3000 + 5000 = 223.60*(d/dt)
2000 = 223.60*(d/dt)
d/dt = 2000 / 223.60
d/dt = 8.94 km/h (approx)
Therefore, the distance between the ships is changing at approximately 8.94 km/h at 4:00 PM.