The coordinates of the point on the curve where the tangent line has a slope of 1/2 are (4, 0).
To find the coordinates of the points on the curve x(t) = 3t^2 + 1 and y(t) = t^3 - 1, where the tangent line has a slope of 1/2, follow these steps:
1. Calculate the derivatives of x(t) and y(t) with respect to t:
dx/dt = 6t
dy/dt = 3t^2
2. The slope of the tangent line is given by dy/dx, so calculate dy/dx using the chain rule:
dy/dx = (dy/dt) / (dx/dt) = (3t^2) / (6t) = t/2
3. Set the slope equal to 1/2 and solve for t:
t/2 = 1/2
t = 1
4. Plug the value of t back into the original equations for x(t) and y(t) to find the coordinates of the point:
x(1) = 3(1)^2 + 1 = 4
y(1) = (1)^3 - 1 = 0
The coordinates of the point on the curve where the tangent line has a slope of 1/2 are (4, 0).
To learn more about coordinates, refer below:
https://brainly.com/question/16634867
#SPJ11
Find the following unknowns about the circle. Round all answers to the nearest tenth.
The given circle has a radius of 11.2,
The diameter of the given circle is 22.4The circumference of the given circle is 70.336 The area of the given circle is 393.88Given radius of the circle (r) = 11.2
The diameter of the given circle = 2*r = 2*11.2 = 22.4
The circumference of the given circle = 2πr = 2*3.14*11.2 = 70.336
The area of the given circle = πr² = 3.14*(11.2)² = 393.88
From the above analysis, we can conclude that the diameter of the circle is 22.4 and the circumference of the circle is 70.336 and the area of the circle is 383.88.
To know more about the circle,
https://brainly.com/question/28162977
#SPJ1
The volume of a paper cone of radius 2. 4cm is 95. 4 cm3. The paper is cut along the slant height from O to AB. The cone is opened to form a sector OAB of a circle with centre O. Calculate the sector angle x°. [The volume, V, of a cone with radius r and height h is V= 1/3 x pi x r^2 x h. ]
The sector angle formed by the cone when it is opened is 54°.
V = 95.4 cm³
r = 2.4 cm
Calculating the height of the cone using the volume formula,
V= 1/3 x π x r² x h
Substituting the values -
95.4 = 1/3 x 3.14 x 2.4² x h
95.4 = 6.03 x h
h = 15.8 cm
Calculating the slant height using the Pythagoras theorem -
l = √(h² + r²)
Substituting the values -
l = √(15.8² + 2.4²)
l = 16
Calculating the curved surface area of the cone -
= πrl
= π(2.4)(16)
= 120.6 cm².
Calculating the sector angle of the sector formed -
The curved surface area of the cone = area of the sector formed by the cone
= 120.6 cm².
Area of a sector in a circle = ∅/360 × πr²,
120.6 = ∅/360 × (3.14)(16²)
120.6 = ∅/360 × 803.84
(120.6)(360) = (∅)(803.84)
43,416/803.84 = ∅
∅ = 54°
Read more about cone on:
https://brainly.com/question/6613758
#SPJ4
I need this problem solved.
The relation has been plotted on the graph where the first quadrant has (1, 2) and (2, 4), while the second quadrant contains (-1, 3) and (-2, 4).
What is a graph?In mathematics, a graph is a visual representation or diagram that shows facts or values in an ordered way.
The relationships between two or more items are frequently represented by the points on a graph.
You can compare various data sets using bar graphs.
In a line graph, the data is represented by tiny dots, and the line that connects them indicates what happens to the data.
So, we have the coordinates:
(-1, 3); (-2, 4); (1, 2); (2, 4)
Now, plot it on the graph as follows:
(Refer to the graph attached below.)
(-1, 3) and (-2, 4) are in the 2nd quadrant, and (1, 2) and (2, 4) are in the 1st quadrant.
Therefore, the relation has been plotted on the graph where the first quadrant has (1, 2) and (2, 4), while the second quadrant contains (-1, 3) and (-2, 4).
Know more about a graph here:
https://brainly.com/question/19040584
#SPJ1
Correct question:
Express the relation (-1, 3); (-2, 4); (1, 2); (2, 4) on the graph.
A grocery store’s earnings in dollars can be modeled by the equation y 5 0. 75x 2 0. 15x, where x represents the number of tomatoes that they sell. If they sell 200 tomatoes in one day, how much money do they earn?
The grocery store's earning income is $30,030 if they sell 200 tomatoes in one day.
We need to find how much the grocery store earns when it sells 200 tomatoes in one day. When The grocery store’s earnings in dollars can be modeled by the equation,
y = 0.75x² + 0.15x
where,
x = number of tomatoes they sell = 200
To find the earnings we need to substitute x in the equation it can be given as,
y = 0.75x² + 0.15x
y = 0.75(200)² + 0.15(200)
y = $30,030
Therefore, the grocery store's earning income is $30,030 if they sell 200 tomatoes in one day.
To learn more about income :
https://brainly.com/question/30157678
#SPJ4
Tara wants to prove that a second pair of corresponding angles from KJN and LJM are congruent.
Determine a second pair of corresponding angles from KJN and LJM that are congruent. Then explain how you know that the two angles are congruent
To determine a second pair of corresponding angles from KJN and LJM that are congruent, we can start by identifying the first pair of corresponding angles.
angle JKN is harmonious to angle LJM. thus, we need to find another brace of corresponding angles that involve these same two angles. One possibility is to look at the perpendicular angles formed by the crossroad of KJ and JM. Angle KJM is perpendicular to angle NJL. therefore, angle KJM in KJN corresponds to angle NJL in LJM. thus, these two angles are harmonious.
We can prove that these two angles are harmonious using the perpendicular angles theorem, which states that perpendicular angles are always harmonious. Since KJ and JM cross at point J, angles KJM and NJL are perpendicular angles and must be harmonious. thus, we've shown that the alternate brace of corresponding angles from KJN and LJM that are harmonious are angle KJM and angle NJL.
Learn more about triangles at
https://brainly.com/question/29193314
#SPJ4
If sin a = -4/5
and sec B =5/3
for a third-quadrant angle a and a first-quadrant angle Ã, find the following
(a)
sin(a + B)
(b)
tan(a + b)
(c) the quadrant containing a + B
O Quadrant I
O Quadrant II
O Quadrant III
O Quadrant IV
Since sin(a) is negative and a is in the third quadrant, we can use the Pythagorean identity to find cos(a):
[tex]cos^2(a) + sin^2(a) = 1[/tex]
[tex]cos^2(a) + (-4/5)^2 = 1[/tex]
[tex]cos^2(a) = 9/25[/tex]
cos(a) = -3/5 (since a is in the third quadrant)
Similarly, since sec(B) = 5/3, we can use the definition of secant to find cos(B):
sec(B) = 1/cos(B) = 5/3
cos(B) = 3/5
(a) To find sin(a + B), we can use the sum formula for sine:
sin(a + B) = sin(a) cos(B) + cos(a) sin(B)
= (-4/5)(3/5) + (-3/5)(4/5)
= -12/25 - 12/25
= -24/25
(b) To find tan(a + B), we can use the sum formula for tangent:
tan(a + B) = (tan(a) + tan(B)) / (1 - tan(a) tan(B))
To find tan(a), we can use the identity: [tex]tan^2(a) + 1 = sec^2(a)[/tex]
[tex]tan^2(a) = sec^2(a) - 1 = (5/3)^2 - 1 = 16/9[/tex]
tan(a) = -4/3 (since a is in the third quadrant)
To find tan(B), we can use the identity: tan(B) = sin(B) / cos(B) = 4/3
Plugging these values into the formula for tan(a + B), we get:
tan(a + B) = (-4/3 + 4/3) / (1 + (-4/3)(4/3))
= 0 / (1 - 16/9)
= 0
(c) To determine the quadrant containing a + B, we need to consider the signs of sin(a + B) and cos(a + B).
From part (a), we know that sin(a + B) is negative. To determine the sign of cos(a + B), we can use the Pythagorean identity:
[tex]sin^2(a + B) + cos^2(a + B) = 1[/tex]
Substituting sin(a + B) = -24/25, we get:
[tex](-24/25)^2 + cos^2(a + B) = 1[/tex]
[tex]cos^2(a + B) = 1 - (-24/25)^2[/tex]
cos(a + B) = ±7/25
Since cos(a + B) is positive in the first and fourth quadrants, and negative in the second and third quadrants, we can conclude that a + B is in the third quadrant, since cos(a + B) is negative and sin(a + B) is negative.
To know more about Pythagorean identity refer here
https://brainly.com/question/10285501#
#SPJ11
The thickness of paper manufactured at a manufacturing company is roughly symmetrical with a mean of 0. 0039 inch and a standard deviation of about 0. 00001 inch. A marketing manager wants to divide the values of the thickness of paper by 0. 0001 to make the values easier to manage.
What are the mean and standard deviation of the thickness of paper after dividing by 0. 0001?
The mean is ?
inch.
The standard deviation is ?
inch
The mean of the thickness of paper is 39 inches, and the standard deviation is 0.1 inches.
To find the mean and standard deviation of the thickness of paper after dividing by 0.0001, you should follow these steps:
1. Divide the mean by 0.0001:
Mean after dividing = 0.0039 inches / 0.0001
Mean after dividing = 39 inches
2. Divide the standard deviation by 0.0001:
Standard deviation after dividing = 0.00001 inches / 0.0001
Standard deviation after dividing = 0.1 inches
The mean of the thickness of paper after dividing by 0.0001 is 39 inches, and the standard deviation is 0.1 inches.
To learn more about Mean and standard deviation
https://brainly.com/question/475676
#SPJ11
Lisa invested money into a bank account. The value of the account after t years can be found using the function f(t)=6320(1.054)t . What is the initial value of the account?
The initial value of the account is: 6320
How to solve compound interest problems?Compound interest is defined as the interest you earn on interest. This can be illustrated by using basic math: if you have $100 and it earns 10% interest each year, you'll have $110 at the end of the first year.
The general formula to find compound interest is:
A = P(1 + r/n)^t
where:
A is final amount
P is initial principal balance
r is interest rate
n is number of times interest applied per time period
t is number of time periods elapsed
We are given the equation as:
f(t) = 6320(1.054)^(t)
Thus, the initial value is 6320
Read more about compound interest at: https://brainly.com/question/24274034
#SPJ1
Solve for length of segr
6 cm
4 cm
b
18 cm
b = [?] cm
If two segments intersect inside
Answer:
12
Step-by-step explanation:
multiply 4x18 then divide by 6
Helppppppppppppppppp
Answer:
it would be at point (6,3)
Step-by-step explanation:
If you were to reflect it over the x-axis you would get (-6,3)
You get a part time job earning $12.50/hr. Tips are $4.60/hr average. Deductions are FICA (7.65%) and Federal Tax Withholding (10%). You work for 18 hours. What is your gross base pay?
If you are earning $12.50/hr in a part-time job, then your gross "base-pay" after deductions is $253.48.
Your base pay is $12.50/hr and you worked for 18 hours, so the base pay would be:
⇒ $12.50/hr × 18 hrs = $225;
The tips are $4.60/hr on average, so the "total-tips" earned would be:
⇒ $4.60/hr × 18 hrs = $82.80;
To calculate the "total-deductions", we first find the total percentage deducted, which is the sum of FICA and Federal Tax Withholding:
⇒ 7.65% + 10% = 17.65%,
To find the total amount deducted, we multiply the total percentage deducted by the "base-pay" + "tips";
⇒ 17.65% x ($225 + $82.80) = $54.32,
So the pay, before any deductions, is the sum of "base-pay" and "tips";
⇒ $225 + $82.80 = $307.80;
To find the "net-pay", we subtract "total-deductions" from pay;
We have,
⇒ $307.80 - $54.82 = $253.48,
Therefore, your gross base pay is $253.48.
Learn more about Pay here
https://brainly.com/question/8789980
#SPJ1
Answer:
add the tips and pay together , this equals 17.10
then multiply by the 18 hours and you’ll get 307.80
Step-by-step explanation:
Marcus estimated the mass of a grain of sugar as 6 x 10-4 gram. Based on that
estimate, about how many grains of sugar are there in a small bag of sugar
that weighs 0. 24 kilogram?
There are 400,000 grains of sugar in a small bag of sugar that weighs 0.24 kilograms.
To find out how many grains of sugar are there in a small bag of sugar that weighs 0.24 kilograms, based on Marcus' estimate, follow these steps:
1. Convert the mass of the bag of sugar from kilograms to grams: 0.24 kg * 1000 g/kg = 240 g.
2. Use Marcus' estimate of the mass of a grain of sugar: 6 x 10^-4 g.
3. Divide the total mass of the bag of sugar by the mass of a single grain of sugar: 240 g / (6 x 10^-4 g/grain).
Now, let's perform the calculation:
240 g / (6 x 10^-4 g/grain) = 240 g / 0.0006 g/grain = 400,000 grains.
To learn more about weight
https://brainly.com/question/15468401
#SPJ11
After a teacher handed out
m packs of notebooks with c notebooks in each pack, he has 13 notebooks left. how many notebooks did he originally have?
The teacher originally had [tex]m*c + 13[/tex] notebooks.
How many notebooks the teacher originally had?If the teacher handed out m packs of notebooks with c notebooks in each pack, then the total number of notebooks that he gave out would be [tex]m*c[/tex].
If he gave out m packs of notebooks with c notebooks in each pack and has [tex]13[/tex] notebooks left, then the total number of notebooks he originally had would be:
[tex]m*c + 13[/tex]
Therefore, the expression for the total number of notebooks originally had by the teacher is [tex]m*c + 13[/tex].
To learn more about such notebook problems:
brainly.com/question/2150889
#SPJ11
Consider the parametric equations
x=cos(t)−sin(t);y=cos(t)+sin(t) 0≤t≤2π
a) Eliminate the parameter t to find a Cartesian equation for the parametric curve.
b) Sketch this parametric curve, indicating with arrows the direction in which the curve is traced.
For two parametric equations, x = cos(t)− sin(t) ; y = cos(t) + sin(t) ; 0≤t≤2π
a) Cartesian equation for the parametric curve is represented by x² + y² = 2.
b) The sketch for this parametric curve, with arrows in the direction of curve tracing is present above figure.
A parametric curve in the x-t plane has the equations x=x(t), y=y(t). The curve associates a point of the plane (x,y) to a value of the parameter t. The rectangular form of the curve can be determined by eliminating the parameter t, i.e. determine the parameter in one equation and Substituting this value in the other equation. We have the following parametric equations,
x = cost - sinty = cos(t)+ sint, 0 ≤ t ≤ 2π
(a) we have to eliminate parameter t to determine a cartesian equation for the parametric curve, use x²+ y² = (cos(t) − sin(t))²+ (cos(t) + sin(t))²
=> x² +y² = cos²t + sin²t - 2cost sint + cos²t + sin²t + 2cost sint
=> x² + y² = 2 ( sin²t + cos²t) = 2
which represents a circle curve centered at the origin and having radius √2.
(b) A sketch of this parametric curve is shown above figure and arrows are used to indicate the direction of curve trace.
For more information about parametric equations, visit :
https://brainly.com/question/30076178
#SPJ4
simplified ration of squares to total shapes is 6:16 for every 3 squares there are how many total shapes there fore the simplified ratio of squares to total shape is what
For every 3 squares, there are 5 total shapes. Therefore, the simplified ratio of squares to total shapes is 3 : 5.
What is the simplified ratio of squares to total shapes?If the unsimplified ratio of squares to total shapes is 6 : 16, then we can express this as 6/16, then,we can simplify this ratio by dividing both the numerator and denominator by 2, giving us 3/8.
This means that for every 3 squares, there are 5 total shapes. Therefore, the simplified ratio of squares to total shapes is 3 : 5.
Full question "Unsimplified ratio of squares to total shapes: 6 : 16. For every 3 squares, there are ____ total shapes; Therefore, the simplified ratio of squares to total shapes is __ : __.
Read more about ratio
brainly.com/question/12024093
#SPJ1
Use the rational expression below to match the variables on the left with their excluded value(s) on the right.
The value of b = 4
How to solveThe function becomes undefined when the denominator goes to 0
3x² - 48 = 0
3x² = 48
x² = 16
x = +/ 4
From the given choices, it's x = 4
Rational expressions that utilize ratios of polynomial expressions are referred to as rational expressions. These can be written in the format p(x)/q(x), where both p(x) and q(x) are polynomials with the constraint that q(x) cannot equal zero.
Read more about Rational expressions here:
https://brainly.com/question/25292194
#SPJ1
The square below has an area of x^ 2 − 12 x + 36 What expression represents the length of one side of the square?
The length of one side of the square is x - 6 units
How to determine the lengthThe formula for calculating the area of a square is expressed as;
A = a²
Such that the a is the length of its side
From the information given, we have that;
Area = x^ 2 − 12 x + 36
solve the quadratic expression, we have that;
x² - 6x - 6x + 36
group in pairs
(x²- 6x) - (6x + 36)
factorize the terms
x(x - 6) - 8(x - 6)
Then, we have;
(x - 6) and (x - 6) units
Learn about squares at: https://brainly.com/question/25092270
#SPJ1
PLS HELP!
Joanna went school supply shopping. She spent $23.25 on notebooks and pencils. Notebooks cost $2.49 each and pencils cost $1.08 each. She bought a total of 15 notebooks and pencils. How many of each did she buy?
Answer: 10 pencils and 5 notebooks.
Step-by-step explanation:
We will create a system of equations using the information given. Let n be equal to the number of notebooks and p be equal to the number of pencils.
She spent $23.25 on notebooks and pencils. Notebooks cost $2.49 each and pencils cost $1.08 each.
$2.49n + $1.08p = $23.25
She bought a total of 15 notebooks and pencils.
n + p = 15
Next, we will solve for p by substituting.
n + p = 15 ➜ n = 15 - p
$2.49n + $1.08p = $23.25
$2.49(15 - p) + $1.08p = $23.25
$37.35 - $2.49p + $1.08p = $23.25
$37.35 - $1.41p = $23.25
-$1.41p = -$14.10
p = 10 pencils
Lastly, we will solve for n by substituting:
n = 15 - p
n = 15 - 10
n = 5
he circumference of an inflated basketball is 29.516 inches. What is the volume of the basketball? Use 3.14 for π. Round final answer to the nearest whole number.
Use 3.14 for π. PLSSSS HELPPP
the volume of the basketball is approximately 490 cubic inches. we can get this answer by using volume formula of volume
what is approximately ?
"Approximately" means almost, but not exactly. It is used to indicate that a value or quantity is very close to the true or exact value, but there may be a small difference or error. In mathematical terms, an approximate value is an estimate or a rounded value that is used
In the given question,
To find the volume of the basketball, we first need to find its radius.
Circumference of a sphere = 2πr
29.516 = 2 * 3.14 * r
r = 29.516 / (2 * 3.14) ≈ 4.7 inches (rounded to one decimal place)
Now, we can use the formula for the volume of a sphere:
Volume of sphere = (4/3) * π * r^3
Volume of basketball = (4/3) * 3.14 * (4.7)^3
Volume of basketball ≈ 490 cubic inches (rounded to the nearest whole number)
Therefore, the volume of the basketball is approximately 490 cubic inches..
To know more about approximately , visit:
https://brainly.com/question/30945002
#SPJ1
A box contains 10 red buttons, 30 green buttons, 40 blue buttons, and 20 yellow buttons. ten buttons were removed, tallied, and then returned to the box. three additional samples were taken in the same way. which sample is the best representation of the buttons in the box?
The best representation of the buttons in the box is the third sample.
To determine which sample is the best representation of the button in the box, we need to examine the samples and compare them to the known ratios of the colors of buttons in the box.
First sample: 6 red, 2 green, 1 blue, 1 yellow (out of 10)
Second sample: 3 red, 3 green, 3 blue, 1 yellow (out of 10)
Third sample: 2 red, 2 green, 5 blue, 1 yellow (out of 10)
To compare the samples to the known ratios, we can calculate the percentage of each color in each sample and compare it to the percentage of each color in the box.
The percentages of each color in the box are:
Red: 10/100 = 10%
Green: 30/100 = 30%
Blue: 40/100 = 40%
Yellow: 20/100 = 20%
First sample:
Red: 6/10 = 60%
Green: 2/10 = 20%
Blue: 1/10 = 10%
Yellow: 1/10 = 10%
Second sample:
Red: 3/10 = 30%
Green: 3/10 = 30%
Blue: 3/10 = 30%
Yellow: 1/10 = 10%
Third sample:
Red: 2/10 = 20%
Green: 2/10 = 20%
Blue: 5/10 = 50%
Yellow: 1/10 = 10%
Based on these percentages, we can see that the third sample is the best representation of the buttons in the box, as it is closest to the known ratios of colors in the box.
The first sample is skewed towards red buttons and away from blue and green buttons, while the second sample has equal percentages of red, green, and blue buttons, which is not reflective of the known ratios in the box.
Learn more about ratio at https://brainly.com/question/13863767
#SPJ11
Would anyone be willing to help me out with a few math questions? I'm up late and could really use the help!
There are ten slips of paper in a box, each numbered 1-10. If Gerard reaches into the box without looking, what is the probability that he will get a number less than 3?
69 ptssssssss
Answer: 1/5
Step-by-step explanation:
There are 10 slips of paper.
The only numbers less than three are 1 and 2
The probability that he will pick up a slip of paper less than three is 2 since only 1 and 2 are less than three.
Therefore the probability is 2/10, and when simplified, it is 1/5.
Therefore the answer is 1/5.
If you have any more questions feel free to ask in the comments! I'd be happy to help!
A consumers group is concerned with the mean cost of dining in a particular restaurant. a random sample of 40 charges (in dollars) per person has a mean charge of $39. 7188 with standard deviation of $3. 5476. is there sufficient evidence to conclude that the mean cost per person exceeds $38. 0
The test statistic is calculated to be 4.05, which is greater than the critical value of 2.704 at a significance level of 0.05, indicating strong evidence to reject the null hypothesis and conclude that the mean cost per person exceeds $38.0.
To test if there is sufficient evidence to conclude that the mean cost per person exceeds $38.0, we can perform a one-sample t-test.
Using the given information, the test statistic is calculated as
t = (39.7188 - 38.0) / (3.5476 / √(40)) = 4.05.
Using a t-table with 39 degrees of freedom (n-1), the p-value is found to be less than 0.01.
Since the p-value is less than the significance level of 0.05, we can reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean cost per person exceeds $38.0.
To know more about Null hypothesis:
https://brainly.com/question/28920252
#SPJ4
Peter gets a part-time job cleaning and maintaining his community's swimming pool and spa. 40 Here are some facts about the pool and spa. There is an outlet for a vacuum halfway along the side of the pool. What is the approximate length the hose should be to reach any part of the pool surface from there? Show your work. Answer Between _and _ft.
The length of the hose to reach any part of the pool surface from there will be 22.36 feet.
How to calculate the length:Length of hose = √(L² + W²)
The pool is 20 feet long and 10 feet wide, the length of the hose needed would be approximately:
Length of hose = √(20² + 10²) = √500 = 22.36 feet
Therefore, Peter would need a vacuum hose that is approximately 22.36 feet long to reach any part of the pool surface from the outlet.
Learn more about length on
https://brainly.com/question/28322552
#SPJ1
A cup has a capacity of 320 ml. It takes 58 cups to fill a bucket and 298 buckets to fill a tank. By rounding to 1 significant figure, estimate the capacity of the tank in litres.
______________________________
Notes: 1L = 1,000ml BUCKET:= 320ml × 58= 18,560mlTANK:= 18,560ml × 298= 5,382,000ml= 5,382,000ml ÷ 1,000= 5,382= ~ 5.4LThe Capacity of The Tank Is Approx. 5.4L_______________________________
At a print shop reams of printer paper are stored in boxes in a closet. Each box contains 12 reams of printer paper. A worker uses 4 reams from 1 of the boxes. Which function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet?
The function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet is y = 12x - 4
Let's start by considering the initial amount of printer paper in the closet before any boxes are used. Since each box contains 12 reams of printer paper, if there are x boxes in the closet, then the total number of reams of paper is given by 12x.
Now, if a worker uses 4 reams from one of the boxes, then the total number of reams of paper remaining in the closet is (12x - 4). If we define y as the total number of reams of paper remaining in the closet, then we have:
y = 12x - 4
This function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet.
As x increases, the total number of reams of paper in the closet increases as well. However, each time a worker uses 4 reams of paper from a box, the total number of reams of paper in the closet decreases by 4.
To learn more about function click on,
https://brainly.com/question/12244031
#SPJ1
A rectangular fish tank needs to hold 500 gallons, and it needs to be two feet deep. The top will be open. A. Find the width and length of the tank that will use the smallest amount of glass. B. The tank will be filled with enough water so that there will be two inches of head space. Find the weight of the water in the tank
The weight of the water in the tank is approximately 3,809 pounds.
A. To find the width and length of the tank that will use the smallest amount of glass, we need to consider the surface area of the tank. Let's use "x" to represent the length and "y" to represent the width. The formula for the surface area of a rectangular tank is:
Surface Area = 2xy + 2xz + 2yz
Since the top of the tank will be open, we can ignore the surface area of the top. We know that the tank needs to hold 500 gallons and be 2 feet deep, so we can use the formula for the volume of a rectangular tank to solve for one of the variables:
Volume = Length x Width x Depth
500 = xy x 2
xy = 250
Now we can substitute this into the surface area formula and simplify:
Surface Area = 2(250) + 2xz + 2yz
Surface Area = 500 + 2xz + 2yz
To minimize the surface area, we need to differentiate this formula with respect to one of the variables and set it equal to zero. Let's differentiate with respect to x:
d(Surface Area)/dx = 2z
Setting this equal to zero, we get:
2z = 0
z = 0
This doesn't make sense, so let's try differentiating with respect to y:
d(Surface Area)/dy = 2z
Setting this equal to zero, we get:
2z = 0
z = 0
Again, this doesn't make sense. We can conclude that the surface area is minimized when x = y, so the tank should be square. Since xy = 250, we can solve for the side length of the square:
x^2 = 250
x ≈ 15.81 feet
So the tank should be approximately 15.81 feet by 15.81 feet to use the smallest amount of glass.
B. The volume of the water in the tank will be:
Volume = Length x Width x Depth
Volume = 15.81 x 15.81 x 1.67
Volume = 397.25 gallons
Since the tank needs to hold 500 gallons with 2 inches of head space, we can find the weight of the water using the formula:
Weight = Volume x Density
The density of water is approximately 8.34 pounds per gallon, so:
Weight = 397.25 x 8.34
Weight ≈ 3,313.69 pounds
So the weight of the water in the tank will be approximately 3,313.69 pounds.
A. To minimize the amount of glass used for the rectangular fish tank, you'll need to create a tank with equal width and length (a square base). Since the tank needs to hold 500 gallons and is 2 feet deep, you can use the formula: Volume = Length × Width × Depth. Convert 500 gallons to cubic feet (1 gallon ≈ 0.1337 cubic feet), so 500 gallons ≈ 66.85 cubic feet.
66.85 = Length × Width × 2
33.425 = Length × Width
Since the length and width are equal, you can solve for one of the dimensions:
Length = Width = √33.425 ≈ 5.78 feet
So, the tank dimensions will be approximately 5.78 feet by 5.78 feet by 2 feet.
B. To find the weight of the water in the tank, first determine the volume of the water. There will be 2 inches of headspace (2 inches ≈ 0.167 feet), so the water depth is 2 - 0.167 = 1.833 feet. The volume of the water is:
Volume = Length × Width × Depth = 5.78 × 5.78 × 1.833 ≈ 61.05 cubic feet
To find the weight of the water, multiply the volume by the weight of water per cubic foot (62.43 lbs/cubic foot):
Weight = 61.05 × 62.43 ≈ 3,809 lbs
learn more about volume of a rectangle here: brainly.com/question/30759574
#SPJ11
Find an equation in the slope-intercept form for the line: slope = 4, y-intercept = 4
Answer:
y=4x+4
Step-by-step explanation:
The slope formula is:
[tex]y=mx+b[/tex]
with m being the slope and b being the y-intercept
Given: slope=4, y-intercept=4
We can substitute the slope and the y-intercept into the question:
y=4x+4
Hope this helps! :)
The equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].
What is the slope?The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).
[tex]\text{m}=\dfrac{\text{y}_2-\text{y}_1}{\text{x}_2-\text{x}_1}[/tex]
[tex]\text{m}=\dfrac{(\text{y}\bar{\text{a}}-\text{y}\bar{\text{a}})}{(\text{x}\bar{\text{a}}-\text{x}\bar{\text{a}})}[/tex]
It is given that:
A line with slope 4 and y-intercept 4.
The linear equation in one variable can be made:
As we know,
The standard equation of the line is:
[tex]\text{y} = \text{mx} + \text{c}[/tex]
Here m is the slope and c is the y-intercept.
[tex]\text{m} = 4[/tex]
[tex]\text{c} = 4[/tex]
[tex]\boxed{\bold{y = 4x + 4}}[/tex]
Thus, the equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].
Learn more about the slope here:
brainly.com/question/3605446
The average first year teacher salary in a certain state is known to be $52,000 with a standard deviation of $1500. A researcher tests this claim by averaging the salaries of 25 first year teachers and finding their average salary to be $52,525. Is there significant evidence to suggest that the claim is wrong at the 5% significance level?
Option C is correct. No, the least value is smaller than the critical value.
Null hypothesis (H₀): The average first-year teacher salary is $52,000.
Alternative hypothesis (Ha): The average first-year teacher salary is not $52,000.
The significance level is 5% (or 0.05), which means we will reject the null hypothesis if the probability of obtaining the observed result is less than 5%.
Calculate the standard error of the mean:
Standard Error = Standard Deviation / √n
where n is the number of samples (n = 25 in this case).
Standard Error = $1500 / √25
= $1500 / 5
= $300
Now, perform the hypothesis test using a t-test since the sample size is relatively small (n < 30) and the population standard deviation is unknown.
t-score = (Sample Mean - Population Mean) / Standard Error
t-score = ($52,525 - $52,000) / $300
t-score = $525 / $300
t-score = 1.75
To find the critical value at a 5% significance level with 24 degrees of freedom (n - 1), we can consult a t-table. At a 5% significance level (two-tailed test), the critical t-value is approximately ±2.064.
Since the calculated t-score (1.75) is not greater than the critical t-value (2.064), we fail to reject the null hypothesis.
Therefore, option C is correct. No, the least value is smaller than the critical value.
To learn more on Statistics click:
https://brainly.com/question/30218856
#SPJ12
Complete question:
The average first year teacher salary in a certain state is known to be $52,000 with a standard deviation of $1500. A researcher tests this claim by averaging the salaries of 25 first year teachers and finding their average salary to be $52,525. Is there significant evidence to suggest that the claim is wrong at the 5% significance level?
A. Yes, the least value is greater than the critical value
B. No, the least value is larger than the critical value
C. No, the least value is smaller than the critical value
D. yes, the least value is smaller than the critical value
Find all solutions of the equation in the interval [0, 2). (Enter your answers as a comma-separated list.)
sec(x) =
2√3
/
3
The whole thing is sec(x) = 2 root 3 and then it’s all over another 3
As an example another question is the same premise but
sec(x) = 2
and then the answer is
x= pi/3 , 5pi/3
So it has to be ordered like that I just don’t understand ty
Therefore, the solutions of the equation in the interval [0, 2) are: x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees). Note that the solutions are listed in ascending order.
How to Solve the Equation?The equation is:
sec(x) = 2√3/3
First, we can find the values of x for which sec(x) = 2√3/3. Recall that sec(x) = 1/cos(x), so we have:
1/cos(x) = 2√3/3
Multiplying both sides by cos(x), we get:
1 = 2√3/3 cos(x)
cos(x) = 3/(2√3) = √3/2
Now, we can use the unit circle to find the solutions of the equation in the interval [0, 2).
cos(x) = √3/2 when x is π/6 or 11π/6 (in radians), or 30° or 330° (in degrees), since these are the angles in the unit circle where the x-coordinate is √3/2.
However, we need to make sure that these solutions are in the interval [0, 2). Since the period of sec(x) is 2π, we can add or subtract 2π to any solution to get another solution. Therefore, we need to find the solutions in the interval [0, 2π) that correspond to the solutions we found above.
π/6 is already in the interval [0, 2π), so it is a solution in the interval [0, 2). To find the other solution in the interval [0, 2), we can add 2π to 11π/6:
11π/6 + 2π = 23π/6
23π/6 is not in the interval [0, 2), so we need to subtract 2π instead:
11π/6 - 2π = -π/6
-π/6 is in the interval [0, 2), so it is also a solution in the interval [0, 2).
Therefore, the solutions of the equation in the interval [0, 2) are:
x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees)
Note that the solutions are listed in ascending order.
Learn more about equation here: https://brainly.com/question/28991103
#SPJ1