(a) The rates of changes of the volume are:
When r = 9 inches, dV/dt = 972π cubic inches per minute
When r = 36 inches, dV/dt = 15,552π cubic inches per minute
(b) The rate of change of the volume relates to the radius through a quadratic function not a constant function.
How to find the rates of changes of the volume?a) We know that the formula for the volume of a sphere is V = (4/3)πr³. We can take the derivative with respect to time t to find the rate of change of volume.
dV/dt = 4πr² (dr/dt)
When r = 9 inches, we have:
dV/dt = 4π(9²)(3) = 972π cubic inches per minute
When r = 36 inches, we have:
dV/dt = 4π(36²)(3) = 15,552π cubic inches per minute
b) The rate of change of the volume of the sphere is not constant even though dr/dt is constant because the rate of change of the volume relates to the radius through a quadratic function not a constant function.
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A line has a slope of
-5 and includes the points (7,p) and (8,4). What is the value of p?
[tex](\stackrel{x_1}{7}~,~\stackrel{y_1}{p})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{p}}}{\underset{\textit{\large run}} {\underset{x_2}{8}-\underset{x_1}{7}}} ~~ = ~~\stackrel{\stackrel{\textit{\small slope}}{\downarrow }}{ -5 }\implies \cfrac{4-p}{1}=-5 \\\\\\ 4-p=-5\implies 4=-5+p\implies 9=p[/tex]
Solve for x. Type your answer as a number
From the figure, the value of x is given as 17 units
How to find the value of x?In mathematics, a ratio shows how many times one number contains another.
The ratio of the sides of the figure is as follows
3x-7/x+5 = 2/1
cross and multiply to have 3x - 7 = 2(x+5)
opening the brackets to have
3x -7 = 2x +10
collecting like terms to have
3x-2x = 10+7
x = 17 units
In conclusion, the value of x is 17 units
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If all three dimensiona of
a pyramid increase by a factor of two, by what factor does the volume of pyramid increase
Answer:
8.
Step-by-step explanation:
That would be by a factor of 2^3 = 8.
the radius of a sphere is increasing at a rate of 4 mm/s. how fast is the volume increasing (in mm3/s) when the diameter is 100 mm? (round your answer to two decimal places.)
The volume of the sphere is increasing at a rate of 209,439.51 mm³/s when the diameter is 100 mm
The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 100 mm? (Round your answer to two decimal places).Formula to calculate the volume of a sphere = (4/3) × π × r³where r = radius of the sphere, π = pi = 3.14, d = diameter of the sphere. The diameter of the sphere, d = 100 mm.So, the radius of the sphere, r = d/2 = 100/2 = 50 mm.
Now, we need to find the rate of change of the volume of the sphere when the radius of the sphere is increasing at a rate of 4 mm/s.We know that the volume of the sphere is given by V = (4/3) × π × r³.We have to differentiate the above formula with respect to time (t).dV/dt = d/dt [(4/3) × π × r³]dV/dt = (4/3) × π × 3r² × dr/dt, substitute r = 50 mm and dr/dt = 4 mm/s in the above equation to find dV/dt.dV/dt = (4/3) × π × 3(50)² × 4dV/dt = 209,439.51 mm³/sTherefore, the volume of the sphere is increasing at a rate of 209,439.51 mm³/s .
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There are plans to install underground pipeline from the lake to the water level in Apache Durham Park what is the approximate length of pipe needed to the nearest meter
To determine the approximate length of the underground pipeline needed from the lake to the water level in Apache Durham Park, we would know the distance between the lake and park, as well as the specific path that the pipeline would take from the lake to the park.
Define the term length?Length is a physical or conceptual measurement of the extent of something from one end to the other.
It refers to the distance between two points or the size of an object or entity in the direction of its longest dimension. In mathematics and geometry, length is a fundamental concept used to describe the size and shape of geometric figures and objects.
It is measured in units such as meters, feet, inches, or centimeters
Assuming that we have this information, we can use the distance between the two points as the approximate length of the pipeline. To calculate this distance, we can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) represents the coordinates of the lake and (x2, y2) represents the coordinates of Apache Durham Park.
Once we have the distance between the two points, we can round it to the nearest meter to get the approximate length of the pipeline needed.
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We would know the distance between the lake and park, as well as the precise route that the pipeline would travel from the lake to the park, if we knew the approximate length of the underground pipeline required from the lake to the water level in Apache Durham Park.
Define the term length?A gauge of length is one that shows how far something extends from one end to the other.
It describes the separation of two points or the size of an item or entity measured along its longest axis. Length is a basic notion in mathematics and geometry that is used to describe the size and shape of geometric figures and objects.
Its dimensions are expressed in terms of meters, feet, inches, or millimeters.
Assuming we have this knowledge, we can use the distance between the two locations to estimate the pipeline's length. We can use the following algorithm to determine this distance:
[tex]d=\sqrt{(x_{2}-x_{1}) ^{2}+(y_{2}-y_{1}) ^{2} }[/tex]
where [tex](x_{1},y_{1} )[/tex] stands for the lake's coordinates and [tex](x_{2} ,y_{2} )[/tex] for Apache Durham Park's coordinates.
Once we know how far apart the two locations are, we can round it to the closest meter to determine how long the pipeline should be roughly.
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The complete question is as follows:
Hannah has a cone made of steel and a cone made of granite.
• Each cone has a height of 10 centimeters and a radius of 4 centimeteres.
• The density of steel is approximately 7.75 grams per cubic centimeter.
• The density of granite is approximately 2.75 grams per cubic centimeter.
What is the difference, to the nearest gram of the masses of the cones?
As a result, the difference is 837 grammes between the cones' respective masses.
Describe density.
Mass per unit volume is measured using density. It is described as a substance's mass per unit volume. The following is the density formula:
Mass / Volume equals density.
So, The formula m = V, where m is the mass, is the density, and V is the volume, can be used to determine each cone's mass.
The formula below can be used to determine each cone's mass:
- The mass of a steel cone weighs 1298.21 grams or 7.75 grams per cubic centimeter.
- The granite cone weighs 460.76 grams, or 2.75 grams per cubic centimeter.
- Mass of steel cone = 7.75 g/cm³ × 167.55 cm³ = 1298.21 grams
- Mass of granite cone = 2.75 g/cm³ × 167.55 cm³ = 460.76 grams
for difference of masses = 1298.2 grams - 460.76 grams = 837 grams
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Need to find the value of c but confused which trig function to use etc. thanks a bunch!
Therefore, the value of x in each triangle is approximately 5.4 and 11.1, respectively.
What is trigonometry?Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of trigonometric functions such as sine, cosine, and tangent, which are used to relate the angles of a triangle to the lengths of its sides. Trigonometry has a wide range of applications in fields such as physics, engineering, navigation, surveying, and astronomy. For example, trigonometry is used in navigation to calculate the angles between a ship and a lighthouse, or between two ships, in order to determine their relative positions. In engineering, trigonometry is used to calculate the forces acting on structures such as bridges and buildings, and to design and build machines that rely on the principles of motion and energy.
Here,
For the first triangle, we have:
Using the sine ratio,
sin(36) = x / 9
Solving for x, we get:
x = 9 sin(36)
≈ 5.4
Rounding to the nearest tenth, we get:
x ≈ 5.4
For the second triangle, we have:
Using the tangent ratio,
tan(27) = x / 21
Solving for x, we get:
x = 21 tan(27)
≈ 11.1
Rounding to the nearest tenth, we get:
x ≈ 11.1
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PLEASE PLEASE HELP GIVE ME A WELL EXPLAINED ANSWER AND I WILL GIVE YOU 100 POINTS
Before a renovation, a movie theater had 140 seats. After the renovation, the theater has 171 seats. What is the approximate percentage increase of the number of seats in the theater? If necessary, round to the nearest tenth of a percent.
Answer:22.1%
Step-by-step explanation:
Money Magic
How did you use the 3 meters and the “Show Earnings Report” throughout the game?
Factor
F(x)= x^2 + 2x - 35
Answer:
Step-by-step explanation:
The population of Vatican City is approximately 800 people. The population of China is about 1.75 × 106 times greater. What is the approximate population of China?
A.
1.4 × 106
B.
1.4 × 107
C.
1.4 × 108
D.
1.4 × 109
To find the approximate population of China, we need to multiply the population of Vatican City by 1.75 × 106:
Population of China = Population of Vatican City × 1.75 × 106
Population of China = 800 × 1.75 × 106
Population of China = 1.4 × 109
Therefore, the approximate population of China is 1.4 × 109, which is option D.
18. (5) two conducting plates have charge /- 0.0000470 mc and each has area 0.138 m2. what is the strength of the electric field between the plates? m = milli
The magnitude of the electric field between the plates is 3.86 × [tex]10^7[/tex]N/C. The sign indicates the direction of the electric field, which depends on the sign of the charge on the plates.
The strength of the electric field between the plates can be calculated using the formula:
E = σ/ε0
where σ is the surface charge density, ε0 is the permittivity of free space, and E is the electric field strength.
The surface charge density is given by:
σ = Q/A
where Q is the charge on each plate and A is the area of each plate.
Substituting the given values, we get:
σ = ±0.0000470 C / 0.138 [tex]m^2[/tex] = ±0.000341 C/[tex]m^2[/tex]
The permittivity of free space is:
ε0 = 8.85 × [tex]10^{-12[/tex] F/m
Substituting the values, we get:
E = σ/ε0 = ±0.000341 C/[tex]m^2[/tex] / 8.85 × [tex]10^{-12[/tex] F/m = ±3.86 × [tex]10^7[/tex] N/C
The magnitude of the electric field between the plates is 3.86 × [tex]10^7[/tex]N/C. The sign indicates the direction of the electric field, which depends on the sign of the charge on the plates.
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b. Use ratio language to write two different sentences that compare the amount
of ribbon and the number of bouquets.
The ratio of ribbon to bouquets is 2:5, meaning for every 2 meters of ribbon, 5 bouquets can be made. For every 7 bouquets, 3 meters of ribbon are needed, meaning the ratio is 3:7.
What is ratio?A ratio is a comparison of two numbers or quantities that are related to each other in some way. It is the relationship between the sizes of two or more things, expressed as the quotient of one divided by the other.
Ratios can be used to compare different aspects of a set of data, such as the ratio of boys to girls in a classroom, or the ratio of red cars to blue cars on a street. They are also used in many real-world applications, such as finance, engineering, and science.
Ribbon to bouquets ratio is 2:5, which means that for every 2 units of ribbon, there are 5 bouquets.
With a ribbon to bouquet ratio of 1:3, the number of bouquets is three times the amount of ribbon.
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12m ² - 4mn-5n².solve the quadratic equation
The Solutions to 12m ² - 4mn-5n² are:
m = n/3 or m = -5n/3
How did we get these values?To solve the quadratic equation 12m² - 4mn - 5n² = 0, we can use the quadratic formula:
m = (-b ± √(b^2 - 4ac)) / 2a
where a = 12, b = -4n, and c = -5n².
Substituting these values into the formula, we get:
m = (-(-4n) ± √((-4n)^2 - 4(12)(-5n²))) / 2(12)
Simplifying:
m = (4n ± √(16n² + 240n²)) / 24
m = (4n ± √(256n²)) / 24
m = (4n ± 16n) / 24
So the solutions are:
m = n/3 or m = -5n/3
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A worker at an animal shelter recorded data about the adoption of cats in the table shown below.
TIME UNTIL ADOPTION FOR CATS AT SHELTER
Time to Adopt
Age
0000
0
Based on the table, which of these statements are true? Choose all that are correct.
a
Younger than 6 Months
6 Months to 1 Year
1 Year and Older
1 to 7 Days 8 Days to 1 Month More than 1 Month
12
9
4
5
2
6
e
3
6
11
Of all the cats adopted, 34.9% are between 6 months and 1 year old
Of the cats 1 year or older, about 58% took more than 1 month to be adopted.
Of all the cats adopted, about 66% were adopted in a month or less.
C
d Of the cats that took more than a month to adopt, 12.5% were younger than 6 months
Of the cats who took 1 to 7 days to be adopted, 25% were between 6 months and 1 year old.
Based on the table provided, the statement that is true is of all the cats adopted, about 66% were adopted in a month or less.
How to calculate a percentage?To calculate a percentage, you need to divide a part by the whole and then multiply the result by 100. The formula for calculating a percentage is:
Percentage = (Part / Whole) x 100
Alternatively, if you have the percentage and the whole, you can use the following formula to find the part:
Part = (Percentage / 100) x Whole
Based on this, let's prove the true statement:
Total of animals: 58 cats
Total of cats adopted in a month or less: 38
Percentage = 38/ 58 x 100 = 65.5% which can be rounded as 66%
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9. The Star Point Ranger Station and the Twin Pines Ranger Station are 30 miles apart along a straight, mountain road. Each station gets word of a cabin fire in a remote area known as Ben's Hideout. A straight path from Star Point to the fire makes an angle of 34° with the road, while a straight path from Twin Pines makes an angle of 14° with the road. Find the distance, d, of the fire from the road. 34° Star Point Ben's Hideout D d 30 mi 14° Twin Pines 10. In problem 9 we had two expressions that were both equal to d. Use both of your expressions and the value you found for d in problem 7 to check your answers. Explain why they were not exactly equal. Does it matter if this application were real life? Why or why not?
The distance of the fire from the road is d ≈ 6.15 miles + 2.94 miles ≈ 9.09 miles.
The two expressions for d give slightly different values. This is due to the fact that we rounded the values of x and y in our calculations.
How to Solve the Problem using Trigonometry?In problem 9, we can use trigonometry to find the distance, d, of the fire from the road. Let x be the distance from Star Point Ranger Station to the fire, and let y be the distance from Twin Pines Ranger Station to the fire. Then, we have:
tan(34°) = d/x
tan(14°) = d/y
Multiplying both sides of each equation by the respective denominator and simplifying, we get:
d = x tan(34°)
d = y tan(14°)
Since the two expressions for d are both equal, we can set them equal to each other and solve for x and y:
x tan(34°) = y tan(14°)
x = (y tan(14°))/tan(34°)
Substituting the value we found for y in problem 7, which was y = 6.15 miles, we get:
x = (6.15 miles) * tan(14°) / tan(34°) ≈ 2.94 miles
Therefore, the distance of the fire from the road is d ≈ 6.15 miles + 2.94 miles ≈ 9.09 miles.
Now, let's check our answers using both expressions for d:
d = x tan(34°) ≈ 2.94 miles * tan(34°) ≈ 2.94 miles * 0.704 = 2.07 miles
d = y tan(14°) ≈ 6.15 miles * tan(14°) ≈ 6.15 miles * 0.249 = 1.53 miles
As we can see, the two expressions for d give slightly different values. This is due to the fact that we rounded the values of x and y in our calculations. If we use the exact values, we would get slightly different values for d, but they would still be very close.
In real life, it is important to be as accurate as possible when dealing with emergencies such as fires. However, in this case, the difference between the two values of d is relatively small, so it may not have a significant impact on the response to the fire. Nevertheless, it is important to use the most accurate values possible to ensure the safety of those involved.
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The initial amount of money borrowed or deposited?
Answer:
yes
Step-by-step explanation:
I need some help with this question!
You are building a square table. You put a diagonal support on the underside of the tabletop. The diagonal support is 3 meters long. What is a side length of the square table? Round to the nearest tenth, if necessary.
Answer:
2.1
Step-by-step explanation:all side lengths are the same (duh) use 4 and 3 (4 bc there is 4 sides). 4 as the diagonal support
4²=16
3²=9
16-9=7 then use square root for 7 the square root of 7 is 2.6457513111 or 2.1
I need help please!!
The average rate of change of the function f(x) over the interval [-2, -9] is -6.
What is the average rate of change of the function f(x)?To determine the average rate of change of the function f(x) over the interval [-2, -9], we need to find the slope of the secant line that connects the points (-2, f(-2)) and (-9, f(-9)).
We first find the values of f(-2) and f(-9):
f(-2) = (-2)² + 5(-2) + 14 = 4 - 10 + 14 = 8
f(-9) = (-9)² + 5(-9) + 14 = 81 - 45 + 14 = 50
So, the two points are (-2, 8) and (-9, 50).
The slope of the secant line between these two points is:
slope = (f(-9) - f(-2)) / (-9 - (-2)) = (50 - 8) / (-9 + 2) = 42 / -7 = -6
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One night a theater sold 524 movie tickets. An adult's ticket costs $6.50 and a child's ticket cost $3.50. In all, $2881 was taken in. How many of each kind of ticket were sold?
196 children's tickets and 328 adult tickets were sold.
What is a system of equations?
A finite set of equations for which common solutions are sought is referred to as a set of simultaneous equations, often known as a system of equations or an equation system.
Here, we have
Given: One night a theater sold 524 movie tickets. An adult's ticket costs $6.50 and a child's ticket cost $3.50. In all, $2881 was taken in.
Let the amount of the child's ticket be x
Let the amount of adults tickets be y
If the total number of tickets sold is 524 movie tickets, then;
x + y = 524
x = 524 - y...(1)
If an adult ticket cost $6.50 and a child’s ticket cost $3.5 with a total of $2881 in all, then;
3.5x + 6.5 y = 2881
35x + 65y = 28810 ...(2)
Substitute equation 1 into 2:
35x + 65y = 2881
35(542-y) + 65y = 28810
18970 - 35y + 65y = 28810
30y = 9840
y = 328
Put the value of y in equation (1) and we get
x = 524 - 328
x = 196
Hence, 196 children's tickets and 328 adult tickets were sold.
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Drag the expressions into the boxes to correctly complete the table, 25 points
These are the polynomial equations:
A = x^ (1/4) - ∛x + 4√x - 8x + 16
B = 3x² - 5x⁴ + 2x - 12
C = x³ - 7x² + 9x - 5x⁴ - 20
D = x⁵ - 5x⁴ + 4x³ - 3x² + 2x - 1
These are the non-polynomial equations:
E = 4/x⁴ + 3/x³ - 2/x² - 1
F = x⁻⁵ - 5x⁻⁴ + 4x⁻³ - 3x⁻² + 2x⁻¹ - 1
Describe a polynomial?Polynomials are mathematical expressions that only use addition, subtraction, multiplication, and non-negative exponentiation of the variables, along with coefficients (constants that multiply with the variables), coefficients, and constants.
Some of the elements of an equation are coefficients, variables, operators, constants, terms, expressions, and the equal to sign. An equation must always begin with the "=" sign and have terms on both sides.
Let the polynomial equations be represented by the following letters: A, B, C, D, E, and F.
In the equation, we can solve for other values to obtain:
Moreover, a polynomial equation is not an algebraic equation that has a negative exponent or an exponent that is fractional. Thus, negative exponent expressions are not polynomials.
A = x^ (1/4) - ∛x + 4√x - 8x + 16
This polynomial exists.
B = 3x² - 5x⁴ + 2x - 12
This polynomial exists.
C = x³ - 7x² + 9x - 5x⁴ - 20
This polynomial exists.
D = x⁵ - 5x⁴ + 4x³ - 3x² + 2x - 1
It is a polynomial.
E = 4/x⁴ + 3/x³ - 2/x² - 1
It is not a polynomial.
F = x⁻⁵ - 5x⁻⁴ + 4x⁻³ - 3x⁻² + 2x⁻¹ - 1
A polynomial is not what it is.
The polynomials are thus resolved.
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Un vêtements qui coutait initialement 100€ a vu son prix diminuer de 30% une première fois puis une deuxième fois de 20%.Quel pourcentage de réduction correspond à ces deux baisses successives?
Step-by-step explanation:
La première baisse de prix est de 30%, ce qui signifie que le prix du vêtement est maintenant de 70% de son prix initial :
Prix après la première baisse = 100€ - (30% x 100€) = 70€
Ensuite, le prix est réduit à nouveau de 20%. Cela signifie que le prix final est de 80% du prix après la première baisse :
Prix après la deuxième baisse = 70€ - (20% x 70€) = 56€
Le pourcentage de réduction totale correspond donc à la différence entre le prix initial et le prix final, exprimé en pourcentage du prix initial :
Pourcentage de réduction totale = ((100€ - 56€) / 100€) x 100% = 44%
Le vêtement a donc subi une réduction de 44% au total après les deux baisses successives de prix.
The cone and cylinder above have the same radius and height. The volume of the cylinder is 57 cm3. What is the volume of the cone? A. 114 cm3 B. 171 cm3 C. 19 cm3 D. 28.5 cm3
The cone and cylinder above have the same radius and height and the volume of the cone is 171 cubic inches.
Volume of a three-dimensional shape is the space occupied by the shape.
Given that,
Radius and height of the cylinder and the cone are same.
Let r and h be the radius and height of each of the cylinder and the cone.
Volume of the Cone = 1/3 π r² h
Volume of the cone = We have, volume of the cylinder is 57 cubic inches.
1/3 π r² h = 57
Multiplying both sides by 3,
π r² h = 57 × 3
π r² h = 171 cubic inches π r² h
Hence the volume of the cone is 171 cubic inches
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Monica needs 12 lemons per liter of water, to make lemonade for 4 people. What amount of lemons and water does Monica need to prepare lemonade for 30 friends?
Step-by-step explanation:
12 lemons / 4 people * 30 people = 90 lemons
1 liter/ 12 lemons * 90 lemons = 7.5 liters of water
-Solve the equation using the elimination method
-6x+5y=1
6x+4y=-10
To solve this system of equations using the elimination method, we need to eliminate one of the variables. One way to do this is to add the two equations together in a way that will eliminate one of the variables.
First, we can multiply the second equation by -1 to change the sign of all its terms:
-6x + 5y = 1
-6x - 4y = 10
Now we can add the two equations together to eliminate x:
( -6x + 5y ) + ( -6x - 4y ) = 1 + 10
Simplifying the left side and the right side:
-12x + y = 11
Now we have one equation with one variable, y. We can solve for y by isolating it on one side of the equation:
-12x + y = 11
y = 12x + 11
We can substitute this expression for y into either of the original equations to solve for x. Let's use the first equation:
-6x + 5y = 1
-6x + 5(12x + 11) = 1
Simplifying the left side:
54x + 55 = 1
Subtracting 55 from both sides:
54x = -54
Dividing both sides by 54:
x = -1
So the solution to the system of equations is x = -1 and y = 1. We can check this solution by substituting these values into both original equations and verifying that they are true.
Identify the slope and y-intercept of the following equation:
y = 4x + 1
The slope and y-intercept of the following equation are:
Slope = 4.
y-intercept = 1.
What is the slope-intercept form?In Mathematics, the slope-intercept form of the equation of a straight line is represented by this mathematical expression;
y = mx + c
Where:
m represent the gradient, slope, or rate of change.x and y represent the data points.c represent the vertical intercept, y-intercept or initial number.Based on the information provided above, an equation that models the line is represented by this mathematical equation;
y = mx + c
y = 11x + 41
By comparison, we have the following:
mx = 4x
Slope, m = 4.
Initial number or y-intercept, c = 1.
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What is the area of the rhombus shown?
Answer:
Step-by-step explanation:
[tex]168yd^{2}[/tex]
Answer:
168 yd²
Step-by-step explanation:
4 × 7 yd × 12 yd / 2 = 168 yd²
Express the function f(x) = -2(x-4)² + 1 graphically, with a table of values, with a
mapping diagram, and using set notation with integers over the interval 1 ≤x≤7.
The expression of the function using the specified formats are presented as follows;
Please find attached the required graph of the function f(x) = -2·(x - 4)² + 1 created with MS Excel
The table of values for the function can be presented as follows;
[tex]\begin{tabular}{|c|c|c|} \cline{1-2}x& f(x) \\ \cline{1-2}1 & -17\\ \cline{1-2} 2 & -7 \\ \cline{1-2} 3 & -1 \\ \cline{1-2} 4 & 1 \\ \cline{1-2} 5 & -1\\\cline{1-2} 6 & -7 \\ \cline{1-2} 7 & -17 \\ \cline{1-2}\end{tabular}[/tex]
The mapping diagram can be presented as follows;
x [tex]{}[/tex] f(x)
1 → -17
2 → -7
3 → -1
4 → 1
5 → -1
6 → -7
7 → -17
The function expressed using set notation can be presented as follows;
f(x) = {-17, -7, -1, 1, -1, -7, -17} where x ∈ {1, 2, 3, 4, 5, 6, 7}
What is a function?A function is a rule that assigns a unique value for the output for each input value.
The specified function is; f(x) = -2·(x - 4)² + 1
Please find attached the graph of the parabola, created with MS Excel, which is a parabola that opens downwards, with a vertex of (4, 1)
The table of values can be presented as follows;
x | f(x)
-----|------
1 | -17
2 | -7
3 | -1
4 | 1
5 | -1
6 | -7
7 | -17
Mapping diagram:
A mapping diagram is a diagram that illustrates how an element of a set is paired with elements in another set.
The mapping diagram for the function can be made to show how the x-values are mapped to the f(x) values as follows;
1 → -17
2 → -7
3 → -1
4 → 1
5 → -1
6 → -7
7 → -17
Set notation;
The function, f(x) = -2·(x - 4)² + 1 can be expressed using set notation for the interval 1 ≤ x ≤ 7, which is the set of integers between 1 and 7, inclusive, which is presented as follows;
Set of x-values; {1, 2, 3, 4, 5, 6, 7}
Set of f(x) values is therefore; {-17, -7, -1, 1, -1, -7, -17}
Therefore, we get f(x) = -2·(x - 4)² + 1 expressed using set notation can be presented as follows;
f(x) = {-17, -7, -1, 1, -1, -7, -17}, where, x ∈ {1, 2, 3, 4, 5, 6, 7}
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The hour hand of a clock is 4.5 cm long while the minute hand of clock is 5.5 cm long (Take # 3.14) What distance does the tip of the hour hand cover in twelve hours (b) What distance does the tip of the minute hand cover in one (c) Which tip covers a longer distance in (a) and (b) above and by how much
(a) The distance that the tip of the hour hand covers in twelve hours is 28.29 cm.
(b)The distance that the tip of the minute hand covers in one hour is 34.57 cm.
(c) The tip of the minute hand covers a longer distance than the tip of the hour hand by 6.28 cm.
How to find the distance the tip of the hour hand cover in twelve hours?(a) To find the distance that the tip of the hour hand covers in twelve hours, we need to calculate the circumference of the circle that the tip of the hour hand traces in twelve hours. The circumference is given by:
C = 2πr
where r is the length of the hour hand
C = 2 x 22/7 x 4.5
C = 28.29 cm
(b) Also, the distance that the tip of the minute hand covers in one hour is:
C = 2 x 22/7 x 5.5
C = 34.57 cm
(c) We can see that the distance covered by the minute hand in one hour (34.57 cm) is greater than the distance covered by the hour hand in twelve hours (28.29 cm).
difference = 34.57 cm - 28.29 cm = 6.28 cm
Thus, the tip of the minute hand covers a longer distance than the tip of the hour hand by 6.28 cm.
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Landon measures the angles of two triangles. Four of the angles in the two triangles measure 52°, 70°, 44°, and 36°. Which two angles cannot be the angle measures for the two triangles? OA. 100° and 58° OB. 84° and 74° OC. 88° and 68° O D. 92° and 66°
Option D, which has 92° and 66° curves, is the correct choice. The total of these angles is 158°, which is needed for the final two angles.
Are a triangle's edges proportional to 3 to 4 to 5?The triangle with edges in the ratio $3:4:5 is therefore "yes" a right-angled triangle. Note: We can only apply Pythagoras' Theorem to right-angled triangles; it cannot be applied to any other triangles.
Let's add the given angle measures: 52° + 70° + 44° + 36° = 202°. This means that the sum of the measures of the remaining two angles in the two triangles is:
360° - 202° = 158°
As a result, since both options A and B have one angle that is higher than 90 degrees, we can rule them out.
Option C has angles of 88° and 68°, which add up to 156°.
Since the highest sum of two acute angles is 180°, the other two angles would have to have a sum of 204°, which is not feasible.
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