To estimate the error in the hypotenuse length of a right triangle using differentials, we use the trigonometric identity sin(θ) = opposite/hypotenuse, finding the derivative, and then apply the angle error to get an error estimate of
±0.31 cm in the hypotenuse length.
The question asks to use differentials to estimate the error in computing the length of the hypotenuse of a right triangle when one side is 9 cm and the opposite angle is 30° with a possible error of ±1°. To calculate the hypotenuse (c) when dealing with problems involving right triangles, you can use the trigonometric identity sin(θ) = opposite/hypotenuse, which can be rearranged to c = opposite/sin(θ). The differential of c with respect to θ is dc/dθ = -opposite * cos(θ)/sin2(θ). Given that the opposite side is 9 cm and the angle θ is 30°, we can find dc/dθ and then multiply by the possible angle error of ±1° to estimate the error in the hypotenuse length.
Plugging in the given values and converting the angle to radians (since the derivative is taken with respect to radians), we find
dc/dθ ≈ -9 cm × cos(30°)/sin2(30°) ≈ -18 cm.
The possible change in angle is ±1°, which in radians is approximately ±0.0175 radians, thus
Estimated error in hypotenuse ≈ -18 cm × ±0.0175 ≈ ±0.315 cm.
Therefore, the estimated error in the length of the hypotenuse, rounded to two decimal places, is ±0.31 cm.
Lenny uses two pieces of yarn for a craft project. The first piece is 5.89 meters long. The second piece is 2.147 meters long. What is the total length of the two pieces of yarn? Enter your answer in the box. M__
Answer: 8.04 meters
Step-by-step explanation:
length of first piece of yarn = 5.89 meters
length of second piece of yarn = 2.147 meters
Total length of of the two pieces of yarn = length of first piece of yarn + length of second piece of yarn = 5.89 meters + 2.147 meters = 8.037 meters
The rule apply for the addition and subtraction is :
The least precise number present after the decimal point determines the number of significant figures in the answer.
Thus the answer will be 8.04 meters as least precise number 5.89 contains two decimal places.
The base of the building is an equilateral triangle 180 meters on each side. The height of the prism is 25m. Find the height, of the triangular base.
The height of the triangular base is 155.46 m.
To find the height of the equilateral triangle base of the prism, we'll first determine the height of one of its constituent triangles.
The height ( h ) of an equilateral triangle is given by [tex]\( h = \sqrt{3}/2 \times \text{side length} \).[/tex]
Calculate the height of one triangle:
[tex]\[ h = \frac{\sqrt{3}}{2} \times 180 \, \text{m} = \frac{\sqrt{3} \times 180}{2} = \frac{180\sqrt{3}}{2} = 90\sqrt{3} \, \text{m} \][/tex]
Calculate the total height of the prism:
The height of the prism is given as 25 m.
Calculate the total height:
Total height = height of one triangle + height of the prism
[tex]\[ \text{Total height} = 90\sqrt{3} \, \text{m} + 25 \, \text{m} = 90\sqrt{3} + 25 \, \text{m} = 155.46 \, \text{m} \][/tex]
So, the height of the triangular base is 155.46 m
For the box shown, the total area is 94 cm2. determine the value of x
Find the total cost of 3.4 pounds of caramel nut clusters at $8.25 per pound and 5.2 pounds of chocolate covered cashews at $9.35 per pound.
Write an equation of the line in the figure.
List any restrictions on the domain for the equation [tex] \frac{x+9}{x+2} [/tex]
Nancy withdrew $19 from her bank account to pay for markers write in integer to represent this situation
True or false, the decimal form of 3 3/8 is 3.38
If one score is randomly selected from a normal distribution with m = 100 and s = 20, the probability of obtaining a score between x = 90 and x = 100 is p = 0.3085.
a. True
b. False
I need help with this ....!!!!!
In a 4 x 3 factorial design, there are how many levels of the first grouping factor?
find the product. List the units. 8 h $9/h
The product is equivalent to $72.
What is a expression? What is a mathematical equation? What is Equation Modelling?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have a rate of $9 per hour for 8 hours.
Now, for 1 hour, the cost is $9. Therefore, for 8 hours, we can write -
x = 9 x 8
x = $72
Therefore, the product is equivalent to $72.
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The temperature in Fairbanks, Alaska, fell 10 degrees in 2 hours.
Explain why the average temperature change was –5ºF per hour.
The temperature fall can be represented by −10. To find the average change per hour, divide −10 by 2. −10 divided by 2 is −5.
Something greater tha 1/2 yard but less than 5/8 yard??
Unfortunately, arsenic occurs naturally in some ground water†. a mean arsenic level of μ = 8.0 parts per billion (ppb) is considered safe for agricultural use. a well in texas is used to water cotton crops. this well is tested on a regular basis for arsenic. a random sample of 36 tests gave a sample mean of x = 6.8 ppb arsenic, with s = 2.9 ppb. does this information indicate that the mean level of arsenic in this well is less than 8 ppb? use α = 0.01. (a) what is the level of significance? 0.01 state the null and alternate hypotheses. h0: μ = 8 ppb; h1: μ > 8 ppb h0: μ = 8 ppb; h1: μ < 8 ppb h0: μ > 8 ppb; h1: μ = 8 ppb h0: μ < 8 ppb; h1: μ = 8 ppb h0: μ = 8 ppb; h1: μ ≠ 8 ppb (b) what sampling distribution will you use? explain the rationale for your choice of sampling distribution. the standard normal, since the sample size is large and σ is known. the student's t, since the sample size is large and σ is unknown. the student's t, since the sample size is large and σ is known. the standard normal, since the sample size is large and σ is unknown. what is the value of the sample test statistic? (round your answer to three decimal places.) -2.483 (c) estimate the p-value. p-value > 0.250 0.125 < p-value < 0.250 0.050 < p-value < 0.125 0.025 < p-value < 0.050 0.005 < p-value < 0.025 p-value < 0.005 sketch the sampling distribution and show the area corresponding to the p-value. webassign plot webassign plot webassign plot webassign plot (d) based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? are the data statistically significant at level α? at the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. at the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. at the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. at the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) interpret your conclusion in the context of the application. there is sufficient evidence at the 0.01 level to conclude that the mean level of arsenic in the well is less than 8 ppb. there is insufficient evidence at the 0.01 level to conclude that the mean level of arsenic in the well is less than 8 ppb.
Selected from this population. find the probability that the sample mean is greater than 2.1 but less than 2.6.
5x-3 (2x-4)=7x+16. solve for x
124,248-55,679 show me how you get this answer
if a population of scores is normally distributed, has a mean of 45 and a standard deviation of 6, the most extreme 5% of the scores lie beyond the score(s) of _________.
a. 35.13
c. 56.76 and 33.24
b. 45.99
d. 45.99 and 35.13
Final answer:
To find the most extreme 5% of normally distributed scores with a mean of 45 and a standard deviation of 6, we look for scores beyond the z-score cut-offs corresponding to 2.5% and 97.5%. The z-score around 1.96 gives us two scores: 56.76 on the high end and 33.24 on the low end, indicating that answer 'c' is correct.
Explanation:
To determine the most extreme 5% of the scores in a normally distributed population with a mean of 45 and a standard deviation of 6, we need to find the z-score that corresponds to the tails of the distribution where the cumulative area is either less than 2.5% or greater than 97.5% (since 5% is split into two tails of the distribution).
Using a standard normal distribution table or a calculator, the z-score that corresponds to the upper 97.5% is typically about 1.96 (or sometimes 1.645 for a one-tailed test). To find the actual scores corresponding to these z-scores, we use the formula for transforming a z-score into an actual value which is:
X = μ + (z • σ)
For the upper extreme:
X = 45 + (1.96 • 6) = 56.76
For the lower extreme:
X = 45 - (1.96 • 6) = 33.24
Thus, the most extreme 5% of the scores lie beyond scores of 56.76 and 33.24. Therefore, the correct answer is c. 56.76 and 33.24.
19³ x 19³ =
1. 19 *exponent* 3
2. 19 *exponent* 4
3. 19 *exponent* 5
4. 19 *exponent* 6
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Please help!!
I an a factor of 28. the other factor is 4. what number am i
Consider the equation y = 14 – 2x. with y on the vertical axis and x on the horizontal axis, the horizontal intercept of this line is
Suppose in a market for iTune cards demand is Qd = 1500 - 250P, and supply is Qs = 850P. Find the equilibrium P* and Q*.
Bryson hopes to win a three-day vacation in a drawing that is being held at his office. He purchased 40 raffle tickets. There were 500 raffle tickets sold. What is the theoretical probability of Bryson winning the trip?
A-2/25
B-1/5
C-4/5
D-500/40
How many ways are there to select 6 students from a class of 25 to hold six different executive positions on a committee?
Which pair of measurements is not equivalent?
24 feet, 8 yards
24 inches, 2 feet
10 feet, 120 inches
3 miles, 15,480 feet
The pair of measurements that is not equivalent is 3 miles and 15,480 feet. All other pairs are equivalent when converted to the same units. This determines which measurements do not match in the same units.
To determine which pair of measurements is not equivalent, we need to convert them into the same units and compare them:
24 feet, 8 yards: 1 yard = 3 feet, so 8 yards = 8 * 3 = 24 feet. Therefore, 24 feet is equivalent to 8 yards.24 inches, 2 feet: 1 foot = 12 inches, so 2 feet = 2 * 12 = 24 inches. Therefore, 24 inches is equivalent to 2 feet.10 feet, 120 inches: 1 foot = 12 inches, so 120 inches = 120 / 12 = 10 feet. Therefore, 10 feet is equivalent to 120 inches.3 miles, 15,480 feet: 1 mile = 5280 feet, so 3 miles = 3 * 5280 = 15,840 feet. Therefore, 3 miles is not equivalent to 15,480 feet.Thus, the pair of measurements that is not equivalent is 3 miles, 15,480 feet.
Part a determine the correct sketches for v=100cos(ωt+ϕ) versus ωt for ϕ=90∘, 45∘, 0∘, −45∘, and −90∘.
Find the 75th derivative of y = cos(2x).
The 75th derivative of y = cos(2x) can be found by dividing 75 by 4 to find the cycle it falls into due to the repeating pattern of cos and sin. Due to remainder 3 after division, the 75th derivative is equivalent to the 3rd derivative from the cycle, which is 8sin(2x), but scaled by 2 to the power of the number of completed cycles (18). This gives the final answer 2^18 * 8sin(2x).
Explanation:The process of finding the 75th derivative of a function can seem daunting or time-consuming, but with functions like cosines and sines, we can rely on the cycling pattern of these functions to simplify the task. This will significantly decrease the amount of computational effort needed.
When you find the derivative of y = cos(2x), the function cycles in a pattern every four times. The first derivative of y = cos(2x) is -2sin(2x), the second derivative is -4cos(2x), the third derivative is 8sin(2x), and the fourth derivative is 16cos(2x). As you can see, we've returned to the original function (scaled by a factor of 16).
So to find the 75th derivative, divide 75 by 4 to get remainder of 3. This means the 75th derivative is the same as the 3rd derivative, but scaled by 2 to the power of the number of full cycles (which is 18). Hence, the 75th derivative of y = cos(2x) is 2^18 * 8sin(2x).
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After eating a 100 calorie snack mark exercised to burn 100 calories show answer in number line
A flag in the shape of a right triangle is hung over the side of a building as shown below. The total weight of the flag is 250 pounds and it has uniform density. a = 15 and b = 39.
Answer:
(a) The density of flag is [tex]\dfrac{25}{27}[/tex] Pounds per square foot
(b) The weight of strip is [tex]\dfrac{20}{9}(15-h)\Delta h[/tex] Pounds
(c) The work is [tex]\dfrac{20}{9}(15-h)h\Delta h[/tex] foot-pounds
(d) The exact work by roof is 1250 foot-pounds
Step-by-step explanation:
(a) We are given a flag in the shape of a right triangle.
The total weight of flag is 250 pounds and Uniform density.
Base of the flag [tex]=\sqrt{b^2-a^2}[/tex]
[tex]=\sqrt{39^2-15^2}=36[/tex]
Area of the flag [tex]=\dfrac{1}{2}\times 36\times 15 = 270[/tex]
Weight of flag = 250 pounds
[tex]Density =\dfrac{Weight}{Area}=\dfrac{250}{270}=\dfrac{25}{27}[/tex]
Hence, The density of flag is [tex]\dfrac{25}{27}[/tex]
(b) Weight of strip which is h feet below the roof.
Length of strip [tex]=\dfrac{36}{15}(15-h)[/tex]
Width of strip [tex]\Delta h[/tex]
Weight = Density x area
[tex]=\dfrac{25}{27}\times \dfrac{36}{15}(15-h)[/tex]
[tex]=\dfrac{20}{9}(15-h)\Delta h[/tex]
Hence, The weight of strip is [tex]\dfrac{20}{9}(15-h)\Delta h[/tex]
(c) Work slice to move h feet above to the roof
work[tex]=Weight\times displacement[/tex]
[tex]=\dfrac{20}{9}(15-h)\Delta h\times h[/tex]
[tex]=\dfrac{20}{9}(15-h)h\Delta h[/tex]
Hence, The work is [tex]\dfrac{20}{9}(15-h)h\Delta h[/tex] foot-pounds
(d) Exact work on the roof by hanging flag
[tex]W=\int_0^{15}\dfrac{20}{9}(15-h)hd h[/tex]
[tex]W=\dfrac{20}{9}(\dfrac{15h^2}{2}-\dfrac{h^3}{3})|_0^{15}[/tex]
[tex]W=1250-0[/tex]
[tex]W=1250[/tex]
Hence, The exact work by roof is 1250 foot-pounds
Final answer:
The weight of the flag is distributed uniformly across its area, with the entire 250 pounds concentrated at the centroid point.
Explanation:
To find the centroid of a right triangle, we can use the following formulas:
[tex]\[ x_c = \frac{b}{3} \][/tex]
[tex]\[ y_c = \frac{a}{3} \][/tex]
Where:
[tex]- \( x_c \)[/tex] is the x-coordinate of the centroid.
[tex]- \( y_c \)[/tex] is the y-coordinate of the centroid.
[tex]- \( a \)[/tex] is the length of the side perpendicular to the base (height).
[tex]- \( b \)[/tex] is the length of the base.
Given \( a = 15 \) and \( b = 39 \), we can calculate the centroid as follows:
[tex]\[ x_c = \frac{39}{3} = 13 \][/tex]
[tex]\[ y_c = \frac{15}{3} = 5 \][/tex]
So, the centroid of the triangle is located at the point (13, 5).
Now, regarding the weight distribution, if the flag has uniform density, the center of mass (or centroid) would be the point where the total weight is concentrated. Since the total weight of the flag is 250 pounds, and the centroid is the point of concentration of mass, the entire weight of the flag, 250 pounds, can be considered to act at the centroid.
Therefore, the weight of the flag is distributed uniformly across its area, with the entire 250 pounds concentrated at the centroid point.