To determine the intersection points of the helix and the sphere, we substitute the helix's parametric expressions into the sphere's equation, simplify, and solve for t, resulting in two points of intersection upon further substitution back into the helix's equation.
Explanation:The question asks at what points the helix r(t) = (sin t, cos t, t) intersects the sphere x2 + y2 + z2 = 65. To find the intersections, we substitute the parametric equations of the helix into the equation of the sphere. Thus, we get (sin2t) + (cos2t) + t2 = 65. Using the Pythagorean identity sin2t + cos2t = 1, the equation simplifies to 1 + t2 = 65, which further simplifies to t2 = 64. Solving for t, we find t = ±8. Thus, the helix intersects the sphere at the points generated by these t values, which can be found by substituting t back into the helix equations, resulting in (sin(8), cos(8), 8) and (sin(-8), cos(-8), -8), with approximate numerical values after calculations.
One serving of granola provides 4% of the protein you need daily. You must get the remaining 48 grams from other sources. How many grams of protein do you need daily?
The formula for finding the circumference of a circle is C=2πrC=2πr . Rearrange the formula to solve for the radius (r).
If the total price paid for an item is $20.25 and the cost of the item before sales tax is $18.90, what is the rate of sales tax?
20.25 -18.90 = 1.35
1.35 / 18.90 = 0.071428571
= 7.14%
check:
18.90 x 1.34946 Round to 1.35
18.90 + 1.35 = 20.25
Sales tax = 7.14%
If 4 is subtracted from twice a number, the result is 10 less than the number. Write an equation to solve this problem. Use n as the variable to represent the number.
The function f(x)=log[7]x is increasing. True or false
The given function increases as x increases, so the statement is true.
Is the function increasing?Here we have the function:
[tex]f(x) = log_7(x)[/tex]
To see if it is increasing, we can evaluate in two values, 10 and 15, this will serve to see if the function is increasing or not:
[tex]f(10) = log_7(10) = 1.18\\\\f(15) = log_7(15) = 1.39[/tex]
As you can see, when x increases also does f(x), so the function is increasing.
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Product A is an 8 oz bottle of cough medication that sells for $1.36. Product B is a 16oz bottle of cough medication that cost $3.20. Which product has the lower unit price
The third term of an arithmetic sequence is 21, and the eighth term is 56. The first term is _____.
The first term of the given arithmetic sequence is 7. This can be determined by finding the common difference and moving backwards from the third term using this value.
Explanation:The subject of this question is Arithmetic Sequence which is a topic in mathematics. An arithmetic sequence is defined by a common difference (d) between consecutive terms.
To solve this problem, we have to find the common difference first. The difference between the 8th term (56) and the 3rd term (21) is 35. But since there are 5 steps from the 3rd term to the 8th term, each step (or the common difference) must be 35 ÷ 5 = 7.
Now that we have the common difference (7), we can reverse it from the 3rd term (21) back to the 1st term which gives us: 21 - 2(7) = 7. So, the first term is 7.
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Jason deposits $5,000 in a bank account that will pay him 4% simple interest annually. If Jason deposits no more than the initial 5,000 , how much money will be in the account at the end of 5 years ?
Valerie predicts she will be able to hold her breath for 31 second while underwater. If she actually held her breath for 34 seconds, what was Valerie's percent error? Round your answer to the nearest tenth of a percent.
If she actually held her breath for 34 seconds. Then the percentage error will be 8.82%.
What is the percentage error?The amount of error is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.
Valerie predicts she will be able to hold her breath for 31 seconds while underwater.
If she actually held her breath for 34 seconds.
Then the percentage error will be
[tex]\rm error \ \% = \dfrac{34-31}{34} \times 100\\\\\\error \ \% = \dfrac{3}{34} \times 100\\\\\\error \ \% = 8.82 \%[/tex]
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a commercial jet airplane climbs at takeoff with slope m=4/9. How far in the horizontal direction will the airplane fly to reach am altitude of 20,000 ft above the takeoff point?
The horizontal direction in will the airplane flies to reach an altitude of 20,000 ft above the takeoff point will be 45,000 feet.
What is a linear equation?A relationship between two or more parameters that, when shown on a graph, produces a linear model. The degree of the variable will be one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
A commercial jet airplane climbs at takeoff with slope m = 4/9.
Then the equation of the line will be given as,
y = (4/9)x
The y-intercept is zero.
The horizontal direction will the airplane fly to reach an altitude of 20,000 ft above the takeoff point will be
20,000 = (4/9)x
180,000 = 4x
x = 180,000 / 4
x = 45,000 feet
The horizontal direction in will the airplane flies to reach an altitude of 20,000 ft above the takeoff point will be 45,000 feet.
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The volume of a rectangular prism is b3 + 8b2 + 19b + 12 cubic units, and its height is b + 3 units. The area of the base of the rectangular prism is square units.
Answer:
b^2 + 5b +4
Step-by-step explanation:
What type of graph would have the title, "Monthly Company Profit for the Year?"
bar graph
line plot
line graph
pictograph
Answer:
line graph
Step-by-step explanation:
A ratio is a rate. Sometimes always or never true
Find the value of 4 + (27 - 12·2) ÷ 2.
Answer's are 3.5,5.5 and 19
Which one is it
The value of 4 + (27 - 12·2) ÷ 2 would be 5.5.
What is simplification of an expression?Usually, simplification involves proceeding with the pending operations in the expression.
Like, 5 + 2 is an expression whose simplified form can be obtained by doing the pending addition, which results in 7 as its simplified form.
A Simplification usually involves making the expression simple and easy to use later.
WE follow the rule PEMDAS parenthesis, exponent, multipy, divide, add, substart.
4 + (27 - 12·2) ÷ 2
4 + (27 - 24 ) ÷ 2
4 + (3) ÷ 2
4+3/2, do 3/2=1.5
then add it
4+1.5 = 5.5.
Therefore, the correct answer is 5.5
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Find the taylor polynomial t3(x) for the function f centered at the number
a. f(x) = e^(â3x) sin(2x), a = 0
A tape that records one hour and 30 minutes on each side moves at cm a second through the tape player. How long, in metres, is the tape?
Most teachers enjoy teaching students. Most teachers are college graduates. Assuming these two statements are true, do some college graduates enjoy teaching students?
Answer:
yes some some college graduates enjoy teaching students.
Step-by-step explanation:
This question is based on reasoning.
Most teachers enjoy teaching students.
Most teachers are college graduates.
A syllogism is a form of logical reasoning formed by the joining of two or more premises to arrive at a conclusion.
So, here we can say that yes some some college graduates enjoy teaching students.
Three students simplified the expression 2x- 3(y-2x) +-5times -2y.
Amber -4y + 7y
Butch 4x + 7y
Carl 8x + 7y
Who is correct? Explain the error the other students made
A group of ladies meet for an afternoon tea party. They bring all their cats. In all, there are 26 heads and 84 feet. How many ladies and how many cats are in the room
The scores on an exam are normally distributed, with a mean of 74 and a standard deviation of 7. What percent of the scores are less than 81?The scores on an exam are normally distributed, with a mean of 74 and a standard deviation of 7. What percent of the scores are less than 81?
We are given the following variables:
Normal distribution
u = sample mean = 74
s = standard deviation of the samples = 7
x = sample value = 81
Since we are asked to find percent of the scores less than 81, this means we have to find the proportion using the z-score. The formula for z score is:
z = (x – u) / s
Substituting the given values:
z = (81 – 74) / 7
z = 1
Using the standard normal distribution tables for z, we find that:
P = 0.8413 at z = 1
Therefore this means that 84.13% of scores are less than 81.
The Answer is the two small boxes just got it right For Plato
FAR LEFT FAR RIGHT
Find the volume of the wedge-shaped region contained in the cylinder x2 + y2 = 49, bounded above by the plane z = x and below by the xy-plane.
We have that
Volume of the wedge-shaped region
[tex]W=228.65[/tex]
Attached below is a picture of the wedge-shaped region contained in the cylinder
Bearing in mind this shape, we have that the coordinates identifying the cylinder are as follows
Co-ordinates
[tex]0 \leq r \leq 7,\frac{\pi}{2} \leq 0 \leq \frac{\pi}{2},0 \leq z \leq x[/tex]
Where
[tex]0 \leq z \leq x =rcos \phi[/tex]
Generally the equation for the volume of the wedge-shaped region W is mathematically given as
[tex]\int \int \int_W 1dv=\int^{7}_{0} \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \int^{rcos \phi}_{0}[/tex]
[tex]\int \int \int_W 1dv=\int^{7}_{0} r (\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}rcos\phi dv))dr[/tex]
[tex]\int \int \int_W 1dv=(\frac{(7)^3}{3}+\frac{(0)^3}{3})(sin {\pi/2}+sin{-\pi/2})[/tex]
[tex]\int \int \int_W 1dv=228.67[/tex]
Therefore
Volume of the wedge-shaped region
[tex]W=228.65[/tex]
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Find the area of the shaded region when r = 4 feet and the area of the triangle is: 123√ square feet
Use 1.73 for 3√ and 3.14 for π. Round your answer to two decimal places.
An investigator wants to estimate caffeine consumption in high school students. how many students would be required to estimate the proportion of students who consume coffee? suppose we want the estimate to be within 5% of the true proportion with 95% confidence.
We can solve this problem by referring to the standard probability distribution tables for z.
We are required to find for the number of samples given the proportion (P = 5% = 0.05) and confidence level of 95%. This would give a value of z equivalent to:
z = 1.96
Since the problem states that it should be within the true proportion then p = 0.5
Now we can find for the sample size using the formula:
n = (z^2) p q /E^2
where,
p = 0.5
q = 1 – p = 0.5
E = estimate of 5% = 0.05
Substituting:
n = (1.96^2) 0.5 * 0.5 / 0.05^2
n = 384.16
Around 385students are required.
Compare the 3 in 536 to the 3 in 1,36 7.
Draw number 536 using base ten block?
On a resent business trip it took 7 hours 30 minutes to drive 375 miles how many miles per hour was the average
Answer:
50 miles / hours.
Step-by-step explanation:
Speed may be defined as the distance covered by the object per unit time.
Speed = [tex]\frac{distance}{time}[/tex]
the distance of the business trip = 375 miles.
the time taken by the business trip = 7 hours 30 minutes = 7.5 hrs.
Speed = [tex]\frac{375}{7.5}[/tex]
Speed = 50 miles / hours.
Here, the average of miler per hour is 50 miles / hours.
A businessman has 10 employees. His salary is equal to 6 times the average of employees salaries. If the 11 of them receive a total of $64,000 in one year, what was the business mans salary that year ??
x = business man's salary
y = employee's salary
x + 10y = 64000
x = 6y
6y + 10y = 64000
16y = 64000
y = 64000/16 = 4000
x = 6*4000
x = 24000
business man's salary was $24,000
What is the estimate of 462,809+256,738
((81x^3 y^(-1/3))^1/4)/(x^2 y)^1/4 simplify
Simplify x^2-4x-32/x^2+7x+12
Cos (pi/11)+cos (3pi/11)+cos (5pi/11)+cos (7pi/11)+cos(9pi/11)
Answer:
1/2
Step-by-step explanation: