Answers
The expression for the area under the graph of f(x) = √x as a limit of Riemann sums is A = lim [n -> ∞ ] Ax [f(x₁) + f(x₂) + ... + f(xₙ)]
For a continuous function f(x), the area A of the region S that lies under the graph of the function can be expressed as the limit of the sum of the areas of approximating rectangles. Each rectangle has a width of Ax, where A is the interval over which we want to find the area and x is a point within that interval. The height of each rectangle is f(x).
Using this definition, we can find an expression for the area under the graph of f(x) = √x, where 1 ≤ x ≤ 13. We start by dividing the interval [1, 13] into n subintervals, each of width Ax = (13-1)/n = 12/n. We can label the endpoints of the subintervals as x₁, x₂, ..., xₙ+1, where x₁ = 1 and xₙ+1 = 13.
Next, we approximate the area under the curve using n rectangles. The height of the ith rectangle is f(xi), where xi is any point in the ith subinterval. The width of each rectangle is Ax, so the area of the ith rectangle is f(xi) Ax. The total area of the rectangles is then the sum of the areas of each rectangle:
f(x₁) Ax + f(x₂) Ax + ... + f(xₙ) Ax
We can simplify this expression by factoring out the common factor of Ax:
Ax [f(x₁) + f(x₂) + ... + f(xₙ)]
Taking the limit as n approaches infinity, we obtain:
A = lim [n -> ∞] Ax [f(x₁) + f(x₂) + ... + f(xₙ)]
To evaluate the limit, we need to find a formula for the sum of f(xi) for i = 1 to n.
This is a sum of square roots, which can be approximated using numerical methods or evaluated exactly using calculus techniques such as the definite integral.
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Answer:
[tex]\displaystyle \lim_{n \to \infty}\sum^n_{i=1}\left(\dfrac{12}{n}\right)\sqrt[7]{1+\dfrac{12i}{n}}[/tex]
Step-by-step explanation:
The Riemann sum is a method by which we can approximate the area under a curve using a series of rectangles.
Definite Integral Notation (Reimann Sum)The area under the curve of f(x) on the interval [a, b] is represented by:
[tex]\boxed{\begin{minipage}{6cm}$\displaystyle \int^b_af(x)\; \text{d}x=\lim_{n \to \infty}\sum^n_{i=1}f(x_i) \cdot \Delta x$\\\\\\where $\Delta x=\dfrac{b-a}{n}$ and $x_i=a+\Delta x \cdot i&\\\end{minipage}}[/tex]
Δx is the width of each rectangle.
[tex]f(x_i)[/tex] is the height of each rectangle.
[tex]x_i[/tex] is the right endpoint of each rectangle.
Therefore:
[tex]\begin{aligned}\displaystyle \int^b_af(x)\; \text{d}x&=\lim_{n \to \infty}\sum^n_{i=1}f(x_i) \cdot \Delta x\\\\&=\lim_{n \to \infty}\sum^n_{i=1}f\left(a+\left(\dfrac{b-a}{n}\right)i\right) \cdot \left(\dfrac{b-a}{n}\right)\end{aligned}[/tex]
The given interval is [1, 13]. Therefore, a = 1 and b = 13:
[tex]\implies \Delta x=\dfrac{13-1}{n}=\dfrac{12}{n}[/tex]
As f(x) = ⁷√x then:
[tex]\implies f(x_i)=f(a+i \Delta x)=\sqrt[7]{a+\Delta x \cdot i}=\sqrt[7]{1+\dfrac{12i}{n}}[/tex]
Substituting into the summation formula:
[tex]\begin{aligned}\displaystyle \int^{13}_1 \sqrt[7]{x}\; \text{d}x&=\lim_{n \to \infty}\sum^n_{i=1}\sqrt[7]{1+\left(\dfrac{13-1}{n}\right)i} \cdot \left(\dfrac{13-1}{n}\right)\\\\&=\lim_{n \to \infty}\sum^n_{i=1}\left(\dfrac{12}{n}\right)\sqrt[7]{1+\dfrac{12i}{n}}\\\\\end{aligned}[/tex]
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Related Questions
Solve for radius and diameter then what is the area C=226.08in
Answers
The radius of the circle is 35.98 in, the diameter is 71.96 in. and the area of the circle is 4066.98 sq. in.
Here, the circumference of circle is C=226.08 in
We know that the formula for the circumference of circle is C = 2πr
Using this formula we find the value of radius 'r',
C = 2πr
226.08 = 2πr
r = 226.08 / 2π
r = 35.98 in
And the diameter of the circle would be,
d = 2r
d = 2 × 35.98
d = 71.96 in.
And the area of the circle would be,
A = π × r²
A = π × 35.98²
A = 4066.98 sq. in.
Therefore, the required area is 4066.98 sq. in.
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The complete question is:
Solve for radius and diameter then what is the area of the circle if the circumference of circle is C=226.08 in.
2. A set of 120 test scores are normally distributed
with a mean of 82 and a standard deviation of
5.
- The price of a gallon of regular gasoline at 75
a) What percent of the scores are between 72
and 872
b) What is the probability that a score is greater
than 77?
c) What is the probability that a score is less than
82 or greater than 92?
d) About how many students scored outside two
standard deviations of the mean?
a) What percent of gas stations sell a gallon of
Answers
It should be noted that 81.85% of the scores are between 72 and 87.
The probability that a score is greater than 77 is approximately 0.8413.
How to calculate the valuea) The z-score for 72 is calculated as:
z = (72 - 82) / 5 = -2
The z-score for 87 is calculated as:
z = (87 - 82) / 5 = 1
0.8413 - 0.0228 = 0.8185
So, approximately 81.85% of the scores are between 72 and 87.
b) The z-score for 77 is calculated as:
z = (77 - 82) / 5 = -1
1 - 0.1587 = 0.8413
Therefore, the probability that a score is greater than 77 is approximately 0.8413 (or 84.13%).
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r(t) =
t2 − 36
i − j + ln(t − 1)k
(a)
Find the domain of r.
Answers
The domain of the vector function is : ( -6, -2) ∪ ( -2, 6)
The given vector function can be correctly expressed as:
r(t) = (t-2)i / (t+2) + sintj + ln(36-t²)k²
The domain of the given function can be determined by finding the domain of each respective component.
(t-2)i / (t+2)
Not defined, t = -2
Thus, the domain of the first component of the vector = (-∞, 2) ∪ (2, ∞)
The 2nd component = sin t
No restriction on t,
The 3rd component of the function is ㏑(36 - t²)
we know,
Natural number is defined for +ve numbers,
i.e.
36 - t² > 0
(6 + t) (6 - t) > 0
thus, it satisfies the inequality -6 < t < 6
The third domain = (-6,6)
Hence, the domain of the vector function is : ( -6, -2) ∪ ( -2, 6)
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27 divided by 187 ( so the work please
Answers
Answer:
Answer:
= 6 R 25
= 6 25/27
187 divided by 27 equals
6 with a remainder of 25
Step-by-step explanation:
Use the equation to find the number of correct answers Gilberto needs to buy a new shirt that costs 25$. Show your work
Answers
a) An equation that represents the amount of money that Gilberto
receives during a school year: y = 5 + (0.5) x
b) The number of correct answers Gilberto needs to buy a new shirt that costs $25 = 40
Let us assume that y represents the amount of money and x represents the number of correct answers.
We know that One dollar = 100 cents.
so, 50¢ = 0.5 dollars
From the described informtaion, we can write the equation that represents the amount of money that Gilberto receives during a school year as: y = 5 + (0.5) x
Now we need to find the number of correct answers Gilberto needs to buy a new shirt that costs $25
i.e., for y = 25, we need to find the value of x.
Substitute y = 25 in above equation.
25 = 5 + (0.5) x
We solve this equation for x
(0.5) x = 20
x = 20 / (0.5)
x = 40
This means that he needs to give 40 correct answers.
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The complete question is:
Gilberto's grandfather gives Gilberto $5 and then 50¢ for each math question he answers correctly on his math exams for the year.
a. Write an equation that represents the amount of money that Gilberto
receives during a school year. Explain what the variables and numbers mean.
b. Use the equation to find the number of correct answers Gilberto needs to buy a new shirt that costs $25. Show your work.
A foam cube, 5 in. in width has a mass of 62 g. What is the volume and density
Answers
The volume of a cube is calculated by multiplying the length of one side by itself twice. In this case, the volume of the foam cube is 5 in. x 5 in. x 5 in. = 125 cubic inches.
Density is calculated by dividing the mass of an object by its volume. In this case, the density of the foam cube is 62 g / 125 cubic inches = 0.496 g/cubic inch.
So, the volume of the foam cube is 125 cubic inches and its density is 0.496 g/cubic inch.
MARK ME BRAINLEISTSelect the correct answer.
A glass case is in the shape of a rectangular prism. The volume of the case is cubic foot. How many cubic blocks with a side length of foot would
be required to find the volume of the glass case?
OA 2
OB 3
OC. 4
D. 5
Reset
Answers
From the question that we have, the solution that we have gotten is that 4 cubic blocks with a side length of foot would be required to find the volume of the glass case
How to solve the problemWe have been given that a glass case is in shape of a rectangular prism.
The volume of a single cubic block is:
We have
Volume of block
= (1/5) * (1/5) * (1/5)
= 1/125 cubic feet
Next we will now have to divide the volume of the glass case by the volume of a single cubic block:
V olume of case / Volume of the block =
(4/125) / (1/125)
= 4
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A glass case is in the shape of a rectangular prism. The volume of the case is cubic 4/125 foot. How many cubic blocks with a side length of 1/5 foot would be required to find the volume of the glass case?
Find the area of the rhombus below.
Answers
The area of the rhombus is derived to be 70 square units
How to calculate for the area of the rhombusIn calculating for the area of a rhombus, we multiply the length of its two diagonals and divide the result by two
For the given rhombus, we have diagonals:
5 + 5 = 10 and 7 + 7 = 14
thus;
Area of the rhombus = (10 × 14)/2
Area of the rhombus = 140/2
Area of the rhombus = 70 square units.
Therefore, the area of the rhombus is derived to be 70 square units.
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You are 28-year-old healthy male and interested in purchasing life insurance. Using the table, determine what the annual premium for a Whole Life insurance police, given that the face value of the policy is $162,253. If need be, round your answer to the nearest cent.
Answers
The annual premium for a Whole Life insurance policy, for the healthy male, would be A. $ 2, 881.61
How to find the annual premium ?From the given table, a 28 year old male would have to pay $ 17. 76 for each $ 1,000 of coverage of Whole Life Insurance.
For a policy that is worth $ 162, 253 therefore, the annual premium would therefore be:
= Amount per thousand / 1, 000 x Policy coverage
= 17. 76 / 1, 000 x 162, 253
= 0.01776 x 162, 253
= $ 2, 881.61
In conclusion, the annual premium is $ 2, 881.61 .
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The ratio of the volume of cone A to the volume of cone B is 27:8
Cone A and cone B are mathematically similar.
The surface area of cone A is 297 cm²
Show that the surface area of cone B is 132 cm²
Answers
h_b / h_a = r_b / r_a (1)
The ratio of the volumes of cone A to cone B is 27:8, so we have:
(1/3) * π * r_a^2 * h_a / [(1/3) * π * r_b^2 * h_b] = 27/8
Simplifying this equation using Equation (1), we get:
r_b^2 / r_a^2 = 27/8 * h_b / h_a = 27/8 * r_b / r_a
Multiplying both sides by r_a^2, we get:
r_b^2 = 27/8 * r_a^2
Taking the square root of both sides, we get:
r_b = (3/2) * r_a
Substituting this relationship into the formula for the surface area of cone A, we get:
π * r_a * (r_a + √(h_a^2 + r_a^2))
Substituting r_b = (3/2) * r_a and simplifying, we get:
π * r_a * (2.5 * r_a + √(h_b^2 + (2.5 * r_a)^2)) = 297
Simplifying this equation using Equation (1), we get:
π * r_b * (2.5 * r_b + √(h_a^2 + (2.5 * r_b)^2)) = (27/8) * 297
Simplifying this equation, we get:
π * r_b * (2.5 * r_b + √(h_b^2 + (2.5 * r_b)^2)) = 132
Therefore, the surface area of cone B is 132 cm².
Since the cones are mathematically similar, we know that the ratio of their volumes is equal to the cube of the ratio of their radii:
V_A/V_B = (r_A/r_B)^3 = (27/8)
Therefore,
r_A/r_B = (27/8)^(1/3)
We also know that the surface area of cone A is given by:
S_A = πr_A√(r_A^2 + h_A^2) + πr_A^2
where h_A is the height of cone A.
We don't know the value of h_A, but we can use the fact that the cones are mathematically similar to find the ratio of their heights:
h_A/h_B = r_A/r_B
h_B = (r_B/r_A)h_A = (8/27)h_A
Now we can use the fact that the ratio of the surface areas of two similar cones is equal to the square of the ratio of their heights:
S_A/S_B = (h_A/h_B)^2 = (27/8)^2
Substituting the given value of S_A and solving for S_B:
297/S_B = (27/8)^2
S_B = 132 cm²
A basket contains 3 green, 4 yellow, 5 blue and 6 red tickets, one of which is randomly selected.
Giving your answer in its simplest form, what's the theoretical probability of not drawing a yellow ticket?
Probability of not drawing a yellow ticket =
Again using theoretical probability, determine the chance of getting a green or red ticket. Give your answer as a fraction in its simplest form.
Probability of drawing a green or red ticket =
Answers
The theoretical probability of drawing a green or red ticket is 1/2 or 0.5.
We have,
There are a total of 3 + 4 + 5 + 6 = 18 tickets in the basket.
The probability of drawing a yellow ticket.
= 4/18
Since there are 4 yellow tickets out of 18 total tickets.
The probability of not drawing a yellow ticket is equal to the probability of drawing a green, blue, or red ticket.
There are 3 + 5 + 6 = 14 tickets that are not yellow.
The probability of not drawing a yellow ticket.
= 14/18
= 7/9
To find the probability of drawing a green or red ticket, we need to add the probabilities of drawing a green ticket and a red ticket.
The probability of drawing a green ticket.
= 3/18
The probability of drawing a red ticket.
= 6/18
Now,
The probability of drawing a green or red ticket is
= 3/18 + 6/18
= 9/18
= 1/2
Thus,
The theoretical probability of drawing a green or red ticket is 1/2 or 0.5.
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1) m2 x m2
2) w3 x w2
3) 5k3 x 4k4 x k
4) A rectangle and square have the same area.
The rectangle has length 27x and width 3x.
Find the length of each side of the square.
Find the length of each side of the square.
5) Fully simplify the following expression
28x3 ÷ 4x
6) Simplify fully:
To multiply letters e.g., xy you must include the multiplication symbol. E.g., xy should be typed as x*y
fraction numerator 12 c to the power of 8 g to the power of 9 over denominator 4 c to the power of 4 g to the power of 4 end fraction
Answers
m² × m² is equal to m⁴.
w³ × w² is equal to w⁵.
5k³ × 4k⁴ × k is equal to 20k⁸.
The length of each side of the square is 9x.
A simplification of the expression 28x³ ÷ 4x is 7x².
How to calculate the area of a rectangle?In Mathematics and Geometry, the area of a rectangle can be calculated by using the following mathematical equation:
A = LB
Where:
A represent the area of a rectangle.B represent the breadth of a rectangle.L represent the length of a rectangle.By substituting the given parameters into the formula for the area of a rectangle, we have the following;
Area of rectangle = (27x) × (3x)
Area of rectangle = 81x²
Note: Area of square = Area of rectangle
Area of square = (side length)²
81x² = (side length)²
Side length = √(81x²)
Side length = 9x
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Daran has a change jar that contains $0.80 in pennies and nickels. He has 10 more nickels than pennies. How many of each type of coin does he have?
Answers
According to the problem, the value of the coins is $0.80, or 80 cents. We can set up an equation to represent this:
0.01p + 0.05n = 0.80
We also know that Daran has 10 more nickels than pennies:
n = p + 10
We can substitute this expression for "n" into the first equation:
0.01p + 0.05(p + 10) = 0.80
Simplifying:
0.01p + 0.05p + 0.50 = 0.80
0.06p = 0.30
p = 5
Therefore, Daran has 5 pennies. We can use the expression n = p + 10 to find the number of nickels:
n = 5 + 10 = 15
Therefore, Daran has 15 nickels.
geometry please help fast
questions 1 and 2
Answers
find the compound interest on rs 6950 for 3 years if the interest is payable half yearly,the rate for the first two years being 6% p.a and for the third year 9% p.a
Answers
Answer:
Rs 2150.81.
Step-by-step explanation:
The interest rate for the first two years is 6% per annum, which is halved for each six-month period. Therefore, the interest for the first two years would be (6950*(1+(0.03))^8) - 6950 = Rs 1373.53.
For the third year, the interest rate is 9% per annum, which is also halved for each six-month period. Therefore, the interest for the third year would be (6950*(1+0.045))^2 - 6950 = Rs 777.28.
Hence, the total compound interest for three years would be Rs 2150.81.
Simplify √ x + 3√ 3x − 5√ x.
Answers
Answer: -4rootx+3root3x
Step-by-step explanation:
think of the expression under the radicals as like terms, you can only add those that are the exact same 1-5=-4(rootx) can't do anything with the middle
Question content area top
Part 1
The length of a rectangular park is six times its width. The park is surrounded by a 3-foot-wide path. Let x denote the width of the park. Write a quadratic function to represent the total area of the park and the path.
Answers
The quadratic function to represent the total area of the park and the path will be 6(x² + 7x + 6).
Given that:
Width, W = x
Length, L = 6x
Path width, d = 3
The rectangle's area will be given as,
Area of the rectangle = L × W square units
The quadratic function to represent the total area of the park and the path will be given as,
A = (x + 2 · 3) (6x + 2 · 3)
A = (x + 6)(6x + 6)
A = 6(x + 6)(x + 1)
A = 6(x² + 7x + 6)
The quadratic function to represent the total area of the park and the path will be 6(x² + 7x + 6).
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san
3
Mia is the new CEO of a sales company. In her first year, the company made $475,000 in sales. In each of
the following years, her sales are expected to increase at a rate of 1796. If this trend continues, what will be
her company's sales in year 14?
A $6,201,709
B $5,129,458
C $3,656,872
D
$2,935,294
Answers
The company sales in year 14, given the rate of increase and the amount made in sales would be C $3,656,872.
How to find the sales ?In the first year, the sales of the company were $ 475, 000 and this sales is scheduled or predicted to grow at 17 %.
This means that the company sales in year 14 would be :
= Initial sales x ( 1 + rate ) ^ number of years
Initial sales = $ 475, 000
Rate = 17 %
Number of years = 14 years
The company sales in year 14 is:
= 475, 000 x ( 1 + 17 %) ¹⁴
= $3, 656, 872
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14) -1 -10a < -1 or 10 + 3a ≤-5
Answers
The solution to the inequality is:
0 < a or a ≤ -5
How to solve the inequality?
An inequality is a relationship that makes a non-equal comparison between two numbers or other mathematical expressions e.g. 2x > 4.
We have: -1 -10a < -1 or 10 + 3a ≤ -5 and we want to solve for the variable "a".
-1 -10a < -1 or 10 + 3a ≤ -5
-10a < -1 + 1 or 3a ≤ -5 - 10
-10a < 0 or 3a ≤ -15
a < 0/(-10) or a ≤ -15/3
a > 0 or a ≤ -5
Since a a > 0 is the same as 0 < a. We can say:
0 < a or a ≤ -5
Thus, 0 < a ≤ -5
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0
Determine whether the curve is the graph of a function of x.
OYes, it is a function.
O No, it is not a function.
If it is, state the domain and range of the function. (Enter your answers in interval notation. If the curve is not the graph of a function of x, enter DNE.)
domain
range
Answers
The correct statement regarding whether the graph represents a function is given as follows:
No, it is not a function.
When does a graph represents a function?A graph represents a function if it has no vertically aligned points, that is, each value of x is mapped to only one value of y. Vertically aligned points mean that a value of x is mapped to multiple values of y, that is, a single input is mapped to multiple outputs which disqualify the relation as a function.
In the graph, we have that at x = 0 = y-axis, the function has three numeric values, which is a case of vertically aligned points, hence the graph does not represent a function.
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Use the information given in the figure to find the length RU. If applicable, round your answer to the nearest whole number.
Answers
Answer:7
Step-by-step explanation:
use the Pythagoras theorem to find su (40^2-32^2) take the square root =24 use this to solve for ru. (25^2-24^2=49) square root of 49 is
Find the missing angle.
92° 78°
Answers
Answer:
[tex]\huge\boxed{\sf x = 10\°}[/tex]
Step-by-step explanation:
Statement:Angles on a straight line add up to 180 degrees.Solution:According to the statement,
92° + x + 78° = 180°
170° + x = 180°
Subtract 170° from both sidesx = 180° - 170°
x = 10°[tex]\rule[225]{225}{2}[/tex]
Health insurers and the federal government are both putting pressure on hospitals to shorten the average length of stay (LOS) of their patients. In 1996, the average LOS for non-heart patients was 4.6 days. A random sample of 20 hospitals in one state had a mean LOS for non-heart patients in 2000 of 3.8 days and a standard deviation of 1.2 days. How large a sample of hospitals would we need to be 99 percent confident that the sample mean is within 0.5 days of the population mean? Select one: a. 48 b. 7 c. 3 d. 32 e. 96
Answers
The sample of hospitals would we need to be 99 percent confident that the sample mean is within 0.5 days of the population mean is A. 48
How to calculate the sampleWe want to find the sample size (n) that would give us a confidence interval of 0.5 days, so we can rearrange the formula to solve for n:
n = (z*σ/CI)^2
Plugging in the given values:
z = 2.576 (for 99% confidence level)
σ = 1.2 days
CI = 0.5 days
n = (2.576 × 1.2/0.5)²
n ≈ 48
Therefore, the correct option is A.
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12 friends shared 8 small pizzas equally. how much pizzas did each person get?
Answers
Each person would get approximately 0.67 small pizzas when 12 friends share 8 small pizzas equally.
If 12 friends shared 8 small pizzas equally, we can divide the total number of pizzas by the number of friends to determine how much pizza each person gets.
Total number of pizzas = 8 small pizzas
Number of friends = 12
To find how much pizza each person gets, we divide the total number of pizzas by the number of friends;
Pizza per person = Total number of pizzas/Number of friends
Pizza per person = 8 small pizzas / 12 friends
Pizza per person = 0.67 small pizzas
Therefore, each person would get 0.67 small pizzas.
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In the diagram, m < AFB = 78
Which angle is complementary to
Answers
The angle BFC is the angle that is complementary to AFB
What is a complementary angleWhenever the sum of two angles adds up to 90 degrees, they are defined as complementary angles. Take 30 and 60 degrees, for instance; they are complementary angles.
From the definition we have to find the angle when if added to AFB would give us the sum of 90 degrees
AFC is a 90 degree angles
We have AFC = AFB + BFC
90 = 78 + BFC
BFC = 90 - 78
= 12
Hence the angle BFC is the complementary angle
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Jason has three T-shirts of different colors. He shares his T-shirts with his two brothers. How many ways can all three boys trade shirts and not wear the same color twice? A. one B. two C. three D. six
Answers
Answer: 6
Step-by-step explanation:
There are six possible ways that the three boys can initially choose shirts: ABC, ACB, BAC, BCA, CAB, CBA.
NEED HELP WILL GIVE BRAINLIEST AND WILL RATE (Do the one thats C and highlighted, ignore the others) :)
Answers
Step-by-step explanation:
3.3% = .033 in decimal
Use the compounding formula
a) P(t) = 18000 (1 + .033)^t
b) for t =2 this computes to 19207.6 bacteria
Given ⊙K with secants NS and NR, which expression represents PS?
Answers
An expression that represent PS include the following: C. (NP + NQ + QR)/NR - NR.
What is the Tangent Secant Theorem?In Mathematics and Geometry, the Tangent Secant Theorem states that if a secant segment and a tangent segment are drawn to an external point outside a circle, then, the product of the length of the external segment and the secant segment's length would be equal to the square of the tangent segment's length.
By applying the Tangent Secant Theorem to this circle, we have the following:
NP × PS = NQ × QR
PS = (NQ × QR)/NP
In terms of line segment NR, the expression can be rewritten as follows;
(NP + NQ + QR)/NR - NR
Read more on Tangent Secant Theorem here: https://brainly.com/question/26826991
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birth weights for a simple random sample of 195 boys were recorded. the mean weight calculated for this sample was 3.04 kilograms and the standard deviation was 0.71 kilograms
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Answer:
The sample mean birth weight for the 195 boys is 3.04 kilograms and the sample standard deviation is 0.71 kilograms.
Step-by-step explanation:
A pilot wants to fly on a bearing of 60.8. By flying due east he finds that a 59 mph wind blowing from the south puts him on course. Find ground speed of the plane
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Answer:
68
Step-by-step explanation:
We can use vector addition to solve this problem. Let's consider two vectors: the velocity of the plane relative to the ground, which we want to find, and the velocity of the wind relative to the ground, which we know is due south with a magnitude of 59 mph.
Let's call the ground speed of the plane "v" and the angle between the plane's velocity and the due east direction "θ". The bearing of 60.8 is the angle between the plane's velocity and due north, so the angle between the plane's velocity and due east is (90 - 60.8) = 29.2 degrees.
Now we can use trigonometry to find the components of the plane's velocity relative to the ground:
v(cosθ, sinθ) = (v cosθ, v sinθ)
The plane's velocity relative to the wind is due east, so its x-component is v cosθ = 59 mph. Therefore, we can solve for v:
v = 59 / cosθ
v = 59 / cos(29.2) ≈ 67.55 mph
So the ground speed of the plane is approximately 67.55 mph. Round it up to 68.