Final answer:
To find the volume of the solid with a triangular base and equilateral triangular cross-sections, we integrate the area of the cross-sections along the y-axis from 0 to 5 based on the changing side length, which is a linear function of y.
Explanation:
The problem requires finding the volume of a solid with a triangular base and equilateral triangular cross-sections perpendicular to the y-axis. The base of the solid, S, is a right triangle with vertices at (0, 0), (5, 0), and (0, 5). The height of each equilateral triangular cross-section at any point y is the distance from the point (y, 0) to the hypotenuse of the triangular base, which is linear and can be described by the equation x + y = 5. Because cross-sections are equilateral triangles, each side of a cross-section is twice the length of the height. Therefore, the area of a cross-section is ½∙height²∙√3.
As we move along the y-axis from 0 to 5, the length of a side of each cross-section decreases linearly from 5 to 0. We can integrate the area of these cross-sections from y = 0 to y = 5 to find the total volume of the solid. The integral for the calculation of volume is:
∫ (2∙(5-y))² × √3/4 dy from y=0 to y=5
Upon integrating, this gives us the volume of the solid S.
A customer has six (6) $1 bills, three (3) $5 bills, four (4) $10 bills, seven (7) quarters, ten (10) dimes, seven (7) nickels, and nine (9) pennies. The customer buys a pair of shoes for $49.86. Based on the combination of bills and coins the customer has, what are the least number of bills and coins the customer can give the cashier in order to buy the shoes for the exact amount and not require any change back?
distributive property of the product of 127 and 32
Complete the square
An air traffic controller is tracking two planes. To start, Plane A is at an altitude of 3427 feet and Plane B is at an altitude of 5000 feet. Plane A is gaining altitude at 65.75 feet per second and Plane B is gaining altitude at 35.5 feet per second.
How many seconds will pass before the planes are at the same altitude?
What will their altitude be when they're at the same altitude.
Gwen has a £5 note and a £2 coin
A liter of cola costs £1.25
Gwen buys as many liter bottles as she can
How much money will she have left over
She will have £0.75 left over after the purchases.
First, we calculate the total amount of money Gwen has:
£5 (note) + £2 (coin) = £7
Next, we determine how many liter bottles of cola Gwen can buy:
Each bottle costs £1.25
Number of bottles she can buy = £7 ÷ £1.25
= 5 bottles
Now, we compute the total cost of 5 bottles:
5 bottles × £1.25/bottle
= £6.25
Finally, we find out how much money Gwen will have left over:
Total money - Total cost = £7 - £6.25
= £0.75
Gwen will have £0.75 left over after buying as many liter bottles of cola as she can.
Catherine walks her dog 3/4 mile everyday how far does she walk each week
Value of the expression
Find the slope of a line that passes through the points (-3,-1) and (0,-5)
Mr. Small, the store manager for Jay's Appliance, is having a difficult time placing a selling price on a refrigerator that cost $410. Mr. Small knows his boss would like to have a 45% markup based on cost. The selling price should be
Answer:
$594.50
Step-by-step explanation:
1. Divide markup into decimal form
45/100 = .45
2. multiply by cost of Refrigerator
.45 x 410 = $184.50
3. Add markup cost to original Refrigerator cost.
184.50 + 410 = $594.50
if R and S are two points in a plane, the perpendicular bisector of line RS is the set of all points equidistant from R and S. True or false?
Answer: True
Step-by-step explanation:
Melanie connected a brown garden hose, a green garden hose, and a black garden hose to make one long hose. The brown hose is 10.75 feet long, the green hose is 16.4 feet long, and the black hose is 8.5 feet long. What is the farthest distance the one long hose can reach?
Answer:
35.65 feet.
Step-by-step explanation:
We have been given that the brown hose is 10.75 feet long, the green hose is 16.4 feet long, and the black hose is 8.5 feet long. We are asked to find the distance that one long hose can reach.
The length of one long hose will be equal to sum of distances of each hose.
[tex]\text{The length of one long hose}=10.75\text{ ft}+16.4\text{ ft}+8.5\text{ ft}[/tex]
[tex]\text{The length of one long hose}=35.65\text{ ft}[/tex]
Therefore, the one long hose can reach 35.65 feet.
The circumference of circle p is 800 mm, the circumference of circle q is 200 cm, and the circumference of circle r is 4 m. What is the sum of the distance around each circle
Let us use the following conversions:
1 cm = 10 mm
1 meter = 100 cm = 1000 mm
Given:
p = 800 mm
q = 200 cm = 2000 mm
r= 4 m = 4000
X = sum of the distance around each circle
X = p+q+r
X = 800+2000+4000
X = 6800 mm or 680 cm or 6.80 m
Determine the inverse of f(x) = x^3 - x^2 - 2x show steps
Switch the x and y values to find the inverse.
y=x−3x+2
The inverse is given by
x=y−3y+2
Solve for y now:
x(y+2)=y−3
xy+2x=y−3
2x+3=y−xy
2x+3=y(1−x)
2x+31−x=y
The inverse, f−1(x), is given by f−1(x)=2x+31−x.
The function can be graphed using knowledge of asymptotes, invariant points, and intercepts. Prepare a table of values for f(x). Recall that f−1(x) is simply a transformation of(x) over the line y=x, so f−1(x) has a table of values where X and y are inverted relative to f(x).
For example, if the point (2,3) belongs on the graph of f(x), the point (3,2) belongs on f−1(x).
Find the measure of angle y. Round your answer to the nearest hundreth.
You toss a coin a randomly selecte a number from 0 to 9. What is the probability of getting tails and selecting a 9?
A.0.05
B.0.95
C.0.25
D.0
a patient is to take 4 1/4 tablespoons of medicine per day in 5 equally divided doses. how much medicine is to be taken in each dose?
Follow below steps:
A patient is to take 4 1/4 tablespoons of medicine per day in 5 equally divided doses. How much medicine is to be taken in each dose?
Convert 4 1/4 tablespoons to a proper fraction, which is 17/4 tablespoons.
Divide 17/4 by 5 to find out how much medicine is to be taken in each dose. This equals 17/4 ÷ 5 = 17/4 × 1/5 = 17/20 tablespoons.
The school cafeteria is baking cookies for lunch. each student gets 3 cookies with their lunch. if there are 231 children buying lunch, how many cookies do they have to make?
What is the justification for each step in solving the inequality?
−2(x+1)≥3x+8−2(x+1)≥3x+8
Select from the drop-down menus to correctly justify each step.
2nd picture is the dropdown box answers
the area of a square is 225 square inches. find the length of a side of the square.
Three consecutive integers have a sum of 297 . Find the integers.
297 /3 = 99
99-1 = 98
99 +1 = 100
98 + 99 + 100 = 297
the numbers are 98, 99 , 100
Which of the following terms, when added to the given polynomial, will change the end behavior?
y = –2x7 + 5x6 – 24
a)–x8
b)–3x5
c)5x7
d)1,000
e)–300
Answer:
1.A
2.C
Step-by-step explanation:
The end behaviour of the polynomial function y = -2x⁷ + 56x⁶ - 24 will change for a) - x⁸ and c) 5x⁷.
What is the end behavior of a polynomial?A polynomial function's final behavior is how its graph behaves as x gets closer to positive or negative infinity.
The graph's final behavior is determined by a polynomial function's degree and leading coefficient.
Given, a polynomial function y = -2x⁷ + 56x⁶ - 24.
So, the given polynomial function has a degree of 7 and its end behavior will be changed by adding or subtracting terms that are of degree 7 and higher.
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What is the average rate of change of f(x), represented by the graph, over the interval [0,2]?
A: 2
B: 1
C: 0.5
D: -0.5
Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. are the given families of curves orthogonal trajectories of each other? that is, is every curve in one family orthogonal to every curve in the other family? x2 + y2 = ax x2 + y2 = by
Yes, the given curves are orthogonal. A further explanation is below.
Given equation is:
[tex]x^2+y^2=ax[/tex]By differentiating both sides, we get
→ [tex]2x+2yy'=a[/tex]
→ [tex]y'=\frac{a-2x}{2y} = m_1[/tex]
again,
[tex]x^2+y^2=by[/tex]By differentiating both sides, we get
→ [tex]2x+2yy' =by'[/tex]
→ [tex]y' = \frac{-2x}{2y-b}[/tex]
For both curves are orthogonal, we get
→ [tex]m_1 \ m_2 = -1[/tex]
By substituting the values, we get
→ [tex]\frac{(a-2x)}{2y} \ \frac{(-2x)}{2y-b} = -1[/tex]
→ [tex]-2ax +4x^2=-4y^2+2yb[/tex]
→ [tex]4(x^2+y^2)=2ax+2yb[/tex]
Since,
[tex]ax=x^2+y^2[/tex][tex]by=x^2+y^2[/tex]then,
→ [tex]4(x^2+y^2) =2(x^2+y^2)+2(x^2+y^2)[/tex]
→ [tex]4x^2+4y^2=4x^2+4y^2[/tex] (true)
Thus the above response is appropriate.
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tina wants to save money for school. tina invests 1100 in an account that pays an interest of 7.25%. how many years will it take the account to reach 6600?
It would take approximately 19 years for Tina's investment to grow from $1100 to $6600 with an annual interest rate of 7.25% if the interest is compounded annually.
Explanation:This problem deals with the concept of compound interest. To find out how many years it will take for Tina's investment to grow from $1100 to $6600 with an interest rate of 7.25%, we would use the formula for compound interest: A = P(1 + r/n)_(nt), where A is the final amount, P is the principal amount (initial investment), r is the annual interest rate (in decimal form), n is the number of times interest is compounded per time unit (year), t is the time the money is invested for in years.
In Tina's case, she does not make additional contributions, so we assume the interest is compounded once per year (n=1). Our formula becomes A = P*(1 + r)_t. Arranging for t, we get t = log(A/P) / log(1+r).
Using these values: A=$6600, P=$1100, r=7.25/100=0.0725, we can find t = log(6600/1100) / log(1+0.0725). Calculating this, you would get around 19 years.
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Suzanne bought 50 apples at the apple orchard she bought four times as many red apples has green apples how many more red apples and green apples that Suzanne buy
For the function f(x) = −2(x + 3)2 − 1, identify the vertex, domain, and range.
The vertex is (3, −1), the domain is all real numbers, and the range is y ≥ −1.
The vertex is (3, −1), the domain is all real numbers, and the range is y ≤ −1.
The vertex is (−3, −1), the domain is all real numbers, and the range is y ≤ −1.
The vertex is (−3, −1), the domain is all real numbers, and the range is y ≥ −1.
Answer:
C. The vertex is [tex](-3,-1)[/tex], the domain is all real numbers, and the range is [tex]y\leq -1[/tex].
Step-by-step explanation:
We have been given a function [tex]f(x)=-2(x+3)^2-1[/tex]. We are asked to identify the vertex, domain and range of the given function.
We can see that our given parabola is in vertex form [tex]y=a(x-h)^2+k[/tex], where [tex](h,k)[/tex] represents vertex of parabola.
We can rewrite our given equation as:
[tex]f(x)=-2(x-(-3))^2-1[/tex]
Therefore, the vertex of our given parabola would be [tex](-3,-1)[/tex].
We know that parabola is a quadratic function and the domain of a quadratic function is all real numbers.
We know that range of a quadratic function in form [tex]f(x)=a(x-h)^2+k[/tex] is:
[tex]f(x)\leq k[/tex], when [tex]a<0[/tex] and,
[tex]f(x)\geq k[/tex], when [tex]a>0[/tex]
Upon looking at our given function, we can see that [tex]a=-2[/tex], which is less than 0, therefore, the range of our given function would be [tex]y\leq -1[/tex].
The student body of a large university consists of 60% female students. a random sample of 8 students is selected. what is the probability that among the students in the sample at least 7 are female?
The probability of selecting at least 7 female students from the sample of 8 students is 0.2797.
To solve this problem, we'll use the binomial probability formula, which calculates the probability of a certain number of successes (in this case, selecting female students) in a fixed number of trials (the sample size).
Given:
Probability of selecting a female student (success), ( p = 0.60 )
Probability of selecting a male student (failure), ( q = 1 - p = 0.40 )
Sample size, ( n = 8 )
We need to calculate the probability of selecting at least 7 female students from the sample.
Calculate the probability of selecting exactly 7 female students:
[tex]\[ P(X = 7) = \binom{8}{7} \times (0.60)^7 \times (0.40)^{8-7} \][/tex]
[tex]\[ = \frac{8!}{7!(8-7)!} \times (0.60)^7 \times (0.40)^{1} \][/tex]
[tex]\[ = 8 \times 0.60^7 \times 0.40 \][/tex]
[tex]\[ = 8 \times 0.0279936 \times 0.40 \][/tex]
[tex]\[ = 0.1119744 \][/tex]
Calculate the probability of selecting exactly 8 female students:
[tex]\[ P(X = 8) = \binom{8}{8} \times (0.60)^8 \times (0.40)^{8-8} \][/tex]
[tex]\[ = (0.60)^8 \][/tex]
[tex]\[ = 0.60^8 \][/tex]
[tex]\[ = 0.16777216 \][/tex]
Add the probabilities from Step 1 and Step 2 to get the final probability:
[tex]\[ P(X \geq 7) = P(X = 7) + P(X = 8) \][/tex]
[tex]\[ = 0.1119744 + 0.16777216 \][/tex]
[tex]\[ = 0.27974656 \][/tex]
So, the probability of selecting at least 7 female students from the sample of 8 students is 0.2797.
The degree of the Boolean function given by f(x,y,z,w) = xy + yz + zw is........
A student got 34 pencils for school. If she sharpen 16 of the pencils before school what is her ratio of unsharpened pencils to sharpened pencils
34-16 = 18
so 18 are unsharpened
ratio is 18/16 reduced to 9/8
need this solved (3x+2y)(5x-6y)