Answer:
918
Step-by-step explanation:
Use distributive property.
(810) + (88) + (20) = 918
Answer:
918
Step-by-step explanation: hope it was helpful
very confused on how to approach??? pls helpppp
Answer:
The height of the wall is 52.2 feet.
Step-by-step explanation:
Given:
From the figure shown, HT is the height of Carlos, 'V' is the position of mirror.
So, TV = 4 ft, VS = 36 ft, HT = 5.8 ft.
Let the height of the wall be 'x'. So, JS = 'x'
Now, consider the two triangles HTV and JVS
Statements Reasons
1. ∠ HTV ≅ ∠ JSV Right angles are congruent
2. ∠ HVT ≅ ∠ JVS Given in the figure
Therefore, ΔHTV and Δ JVS are similar triangles by AA postulate.
Now, from definition of similar triangles, corresponding sides of two similar triangles are in proportion to each other. Therefore,
[tex]\frac{HT}{JS}=\frac{TV}{VS}=\frac{HV}{JV}[/tex]
Considering the first two pair of fractions, we have:
[tex]\frac{5.8}{x}=\frac{4}{36}\\\\36\times 5.8=4\times x\\\\x=\frac{36\times 5.8}{4}\\\\x=9\times 5.8=52.2\ ft[/tex]
Therefore, the height of the wall is 52.2 feet.
The volume of a cylinder is increasing at a rate of 10π cubic meters per hour.
The height of the cylinder is fixed at 5 meters.
At a certain instant, the volume is 80π cubic meters.
What is the rate of change of the surface area of the cylinder at that instant (in square meters per hour)?
Answer:
13π/2 m²/h
Step-by-step explanation:
Volume of a cylinder is:
V = πr²h
If h is a constant, then taking derivative of V with respect to time:
dV/dt = 2πrh dr/dt
Surface area of a cylinder is:
A = 2πr² + 2πrh
Taking derivative with respect to time:
dA/dt = (4πr + 2πh) dr/dt
Given that dV/dt = 10π, V = 80π, and h = 5, we need to find dA/dt. But first, we need to find r and dr/dt.
V = πr²h
80π = πr² (5)
r = 4
dV/dt = 2πrh dr/dt
10π = 2π (4) (5) dr/dt
dr/dt = 1/4
dA/dt = (4πr + 2πh) dr/dt
dA/dt = (4π (4) + 2π (5)) (1/4)
dA/dt = 13π/2
The surface area of the cylinder is increasing at 13π/2 m²/h.
The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.
Given that,
The volume of the cylinder increase with the rate of 10π cubic meters per hour.
The height of the cylinder is fixed at 5 meters.
At a certain instant, the volume is 80π cubic meters.
We have to determine,
What is the rate of change of the surface area of the cylinder at that instant.
According to the question,
Height of cylinder = 5m
Volume of cylinder increase with rate = 10π cubic meters per hour.
At certain instant volume become = 80π cubic meters per hour.
Volume of cylinder is given as,
V = πr²h
Where h is a constant,
[tex]v = \pi r^{2} h\\\\80\pi = \pi r^{2}. (5)\\\\\frac{80\pi }{5\pi } = r^{2} \\\\r^{2} = 16\\\\r = 4[/tex]
Then, taking derivative of V with respect to time:
[tex]\frac{dv}{dt} = 2\pi rh.\frac{dr}{dt} \\\\[/tex]
Where, dv\dt = 10π, V = 80π, and h = 5,
Then,
[tex]10\pi = 2\pi (4)(5). \frac{dr}{dt} \\\\10\pi = 40\pi \frac{dr}{dt} \\\\\frac{10\pi }{40\pi } = \frac{dr}{dt} \\\\\\\frac{dr}{dt} = \frac{1}{4}[/tex]
Then,
Surface area of a cylinder is define as,
[tex]A = 2\pi r^{2} + 2\pi rh\\\\\ \frac{da}{dt} = (4\pi r + 2\pi h)\frac{dr}{dt} \\\\\frac{da}{dt} = (4\pi (4) + 2\pi (5))\times\frac{1}{4} \\\\\frac{da}{dt} = \frac{26\pi }{4} \\\\\frac{da}{dt} = \frac{13\pi }{2}[/tex]
Hence, The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.
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a hexagonal aquarium is 15.5 on each side of its base and 28.5cm high. whats the area
Answer:
17807cm2
Step-by-step explanation:
Surface area of a hexagonal prism is =( (3√3)/2) a^2 h
= 2.598 * 15.5^2 *28.5
= 2.598* 240.5* 28.5
=17807.34cm2
a particle is moving on the x-axis, where x is in centimeters. it has velocity v(x) in cm/s, when it is at the co-ordinate x, given by v(x)=x^3-2x+3 find the acceleration of the particle when it is at the point x=2. express youre answer as a number rounded to the nearest whole number
Answer:
10cm/s²
Step-by-step explanation:
Acceleration is the change in velocity of an object with respect to time.
Given velocity v(x) to be x³-2x+6
Acceleration = ∆v/∆x
Differentiating the velocity function to get acceleration we have,
Acceleration = dv/dt = 3x²-2
Acceleration of the particle at x = 2 will give;
dv/dt @ x = 2 is 3(2)²-2
= 12-2 = 10cm/s²
The unit price on a 100-pound container of a swimming pool chlorine is $.06 per pound less than the unit price on a 75-pound container. If the 100-pound container costs $55.50 more than the 75-pound container, find the cost of each.
Answer:
100 lb: $24075 lb: $184.50Step-by-step explanation:
Let u represent the unit price of the 100 lb container. Then u+.06 is the unit price of the 75 lb container.
The difference in prices is ...
100u - 75(u+.06) = 55.50
25u -4.50 = 55.50
25u = 60
u = 60/25 = 2.40
The cost of the 100 lb container is 100u = 240. The cost of the 75 lb container is $55.50 less: 240 -55.50 = 184.50.
The 100 lb container costs $240. The 75 lb container costs $184.50.
Express tan(23°−21°) in terms of tangents of 23∘and 21∘ You do NOT need to type in the degree symbol. Be sure to PREVIEW your answer before submitting!
Answer:
tan(23° -21°) = (tan(23°) -tan(21°))/(1 +tan(23°)tan(21°))
Step-by-step explanation:
The formula for the tangent of the difference of angles is ...
tan(a-b) = (tan(a) -tan(b))/(1 +tan(a)tan(b))
Filling in the values a=23° and b=21°, you get the formula shown above.
The expression for tan(23°−21°) in terms of the tangents of 23° and 21° is: tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°).
To express tan(23°−21°) in terms of the tangents of 23° and 21°, we can use the angle subtraction formula for tangent, which is:
tan(α - β) = (tan α - tan β) / (1 + tan α * tan β)
Applying this to tan(23°−21°), we get:
tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°)
We can't directly calculate the values from the given table since 23° and 21° are not listed. However, we understand the formula required to express tan(23°−21°) in terms of the tangents of these two angles.
A spherical balloon holding 35 lbm of air on earth has a diameter of 10 ft. Determine (a) the specific volume, in unit ft3/lbm, and (b) the weight, in lbf, of the air within the balloon.
Answer:
a) Specific volume of the air in balloon is [tex]14.96 ft^3/lbm[/tex]
b)The weight of the air within the balloon is 1,126.09 lbf.
Step-by-step explanation:
Mass of air, m = 35 lbm
Volume of the air = V
Diameter of balloon = d = 10 ft
radius of the balloon = r= 0.5 d = 5 ft
Volume of balloon = V
[tex]V=\frac{4}{3}\pr r^3[/tex]
[tex]V=\frac{4}{3}\times 3.14\times (5 ft)^3[/tex]
Specific volume of the air in balloon = S
[tex]S=\frac{V}{m}=\frac{\frac{4}{3}\times 3.14\times (5 ft)^3}{35 lbm}[/tex]
[tex]S=14.96 ft^3/lbm[/tex]
Specific volume of the air in balloon is [tex]14.96 ft^3/lbm[/tex]
[tex]lbf=32.174 lbm ft/s^2[/tex]
Weight of the air = W
Acceleration due to gravity = [tex]32.174 lbm ft/s^2[/tex]
Weight = m\times g[/tex]
[tex]W=35 lbm\times 32.174 lbm ft/s^2[/tex]
[tex]W=1,126.09 lbf[/tex]
The weight of the air within the balloon is 1,126.09 lbf.
A quality control engineer at a potato chip company tests the bag filling machine by weighing bags of potato chips. Not every bag contains exactly the same weight. But if more than 15% of bags are over-filled then they stop production to fix the machine.
They define over-filled to be more than 1 ounce above the weight on the package. The engineer weighs 100 bags and finds that 21 of them are over-filled.
He plans to test the hypotheses: H0: p = 0.15 versus Ha: p > 0.15 (where p is the true proportion of overfilled bags).
What is the test statistic?
Z = 1.68
Z = -1.68
Z = 4
Z = -1.47
Answer:
Option A) 1.68
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 100
p = 15% = 0.15
Number of bags overfilled , x = 21
First, we design the null and the alternate hypothesis
[tex]H_{0}: p = 0.15\\H_A: p > 0.15[/tex]
Formula:
[tex]\hat{p} = \dfrac{x}{n} = \dfrac{21}{100} = 0.21[/tex]
[tex]z = \dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
Putting values, we get,
[tex]z = \dfrac{0.21-0.15}{\sqrt{\dfrac{0.15(1-0.15)}{100}}}\\\\z = 1.68[/tex]
Thus, the correct answer is
Option A) 1.68
To test the hypothesis H0: p = 0.15 versus Ha: p > 0.15, the test statistic is Z = 1.68.
Explanation:To test the hypothesis H0: p = 0.15 versus Ha: p > 0.15, where p is the true proportion of overfilled bags, we can use the test statistic Z. The formula for the test statistic is Z = (P' - p) / √(p(1-p) / n), where P' is the sample proportion, p is the hypothesized proportion, and n is the sample size.
In this case, the sample proportion P' = 21/100 = 0.21, the hypothesized proportion p = 0.15, and the sample size n = 100.
Plugging these values into the formula, we get Z = (0.21 - 0.15) / √(0.15(1-0.15) / 100) = 1.68.
Therefore, the test statistic is Z = 1.68.
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You and your friend are going to some candy. You eat 3/4 of a box of candy. Your friend eats 1/2 as much as you do. How much of a box of candy does your friend eat?
Answer:
[tex]\frac{3}{8}[/tex] of a box of candy your friend eat.
Step-by-step explanation:
Given:
You and your friend are going to some candy.
You eat 3/4 of a box of candy.
Your friend eats 1/2 as much as you do.
Now, to find that how much of a box of candy does your friend eat.
Quantity of a box of candy you eat = [tex]\frac{3}{4}[/tex]
Quantity of candy your friend eat = [tex]\frac{1}{2} \ of\ \frac{3}{4}[/tex]
Now, to get how much the candy your friend eat we multiply 1/2 and 3/4:
= [tex]\frac{1}{2} \times \frac{3}{4}[/tex]
= [tex]\frac{3}{8}[/tex] .
Therefore, [tex]\frac{3}{8}[/tex] of a box of candy your friend eat.
A number that indicates the degree and direction of the relationship between variables is called a(n) _____.
Answer: The answer is Correlation because its a connection between 2 or more things, or interdependence of variable quantities. Plus its in the definition, it says the process of establishing a relationship or connection between 2 or more measures.
Peaceful Travel Agency offers vacation packages. Each vacation package includes a city, a month, and an airline. The agency has cities, month, and airlines to choose from. How many different vacation packages do they offer?
Answer:12 packages.
Step-by-step explanation:
The agency has 2 cities,1 month and 6 airlines to choose from.(completion of the question)
Total package of the agency = 2cities×1 month × 6 airlines=12 packages.
3. Using techniques from Calculus, show directly that the maximum value of a 1-D Gaussian distribution occurs at the point x = μ.
Answer:
For a scaler variable, the Gaussian distribution has a probability density function of
p(x |µ, σ² ) = N(x; µ, σ² ) = 1 / 2π×[tex]e^{\frac{-(x-u)^{2}}{2s^{2} } }[/tex]
The term will have a maximum value at the top of the slope of the 1-D Gaussian distribution curve that is when exp(0) =1 or when x = µ
Step-by-step explanation:
Gaussian distributions have similar shape, with the mean controlling the location and the variance controls the dispersion
From the graph of the probability distribution function it is seen that the the peak is the point at which the slope = 0, where µ = 0 and σ² = 1 then solution for the peak = exponential function = 0 or x = µ
7. In which reaction is mass converted to energy by the process of fission?
When dragons on planet Pern lay eggs, the eggs are either green or yellow. The biologists have observed over the years that 32% of the eggs are yellow, and the rest green. Next spring the lead scientist has permission to randomly select 60 of the dragon eggs to incubate. Consider all the possible samples of 60 dragon eggs.What is the usual number of yellow eggs in samples of 60 eggs? (Give answers as SENSIBLE whole numbers.)minimum usual number of yellow eggs =maximum usual number of yellow eggs
Answer:
19
Step-by-step explanation:
If the scientist chooses 60 random eggs then the overall probability of selecting yellow eggs is 32%. So, there's a chance of he randomly collecting 60 eggs out of which 19 could be yellow.
(60/100) x 32 ~ 19...
The usual number of yellow dragon eggs expected in a sample of 60, given a 32% probability, is 19 when rounded to the nearest whole number.
The question at hand involves finding the expected number of yellow eggs in a sample size of 60, given a known probability of a yellow egg occurrence.
Since we know that 32% of the dragon eggs on planet Pern are yellow, we can calculate the usual number of yellow eggs in the samples by multiplying the sample size (60 eggs) by the probability of getting a yellow egg (0.32).
Therefore, the expected or usual number of yellow eggs in the sample would be 60 × 0.32 = 19.2.
Considering that we cannot have a fraction of an egg, the usual number of yellow eggs in samples of 60 eggs would typically be rounded to the nearest whole number, which is 19.
On average, employees at a particular company historically missed 4 days of work per year. Then there was a management change. Now the average is 7 days. What is the percent of increase in the average number of missed days?
Answer:
75% increase
Step-by-step explanation:
Given:
Number of missed days of work before the management change = 4
Number of missed days of work after the management change = 7
So, increase in the number of missed days of work is given by subtracting the old number from the new number and is equal to:
Increase in missed number of days = 7 - 4 = 3 days
Now, the percent of this increase is calculated by dividing the increase by the old number and then multiplying the result by 100.
Percent of increase is given as:
[tex]Percent\ of\ increase=\frac{Increased\ value}{Old\ value}\times 100\\\\Percent\ of\ increase=\frac{3}{4}\times 100\\\\Percent\ of\ increase=0.75\times 100\\\\Percent\ of\ increase=75\%[/tex]
So, there is a 75% increase in the number of missed days of work after the change in the management.
How many kilocalories are in a chicken quesadilla that has 28 grams of protein, 40 grams of carbohydrate, and 30 grams of fat? Group of answer choices
Final answer:
The chicken quesadilla contains a total of 542 kilocalories, which is derived by summing the caloric contributions from protein, carbohydrate, and fat.
Explanation:
To calculate the number of kilocalories in a chicken quesadilla with 28 grams of protein, 40 grams of carbohydrate, and 30 grams of fat, we use the known caloric values for each macronutrient. Protein and carbohydrate each have 4 Calories per gram, while fat has 9 Calories per gram.
Protein: 28 grams × 4 Calories/gram = 112 Calories
Carbohydrate: 40 grams × 4 Calories/gram = 160 Calories
Fat: 30 grams × 9 Calories/gram = 270 Calories
Adding up these amounts, we have:
112 Calories (from protein) + 160 Calories (from carbohydrate) + 270 Calories (from fat) = 542 Calories
To convert Calories into kilocalories, we remember that 1 kilocalorie = 1 Calorie. Therefore, the chicken quesadilla contains 542 kilocalories.
Last spring Janessa made 30 to one cup serving of lemonade for a lemonade stand this year she will make two times the amount I live in a that she made last year how many gallons of lemonade will Janessa make this year
Answer:
3.75 gallons
Step-by-step explanation:
The 5th grade is getting a special lunch to celebrate the end of first grading period.The cafeteria manager is planning to buy 0.3 pound of turkey for each student. If turkey is on sale for $0.79 per pound, what will it cost to give turkey to 100 students.
Final answer:
To calculate the cost, multiply the amount of turkey per student (0.3 pounds) by the number of students (100), then multiply the total pounds needed (30) by the cost per pound ($0.79), resulting in a total cost of $23.70 for turkey for 100 students.
Explanation:
The question asks us to calculate the cost of providing turkey to 100 students if each student requires 0.3 pounds of turkey, and turkey costs $0.79 per pound. To find the total amount of turkey needed, we multiply the amount each student gets by the total number of students: 0.3 pounds/student × 100 students. This calculation results in 30 pounds of turkey required for 100 students.
Next, to determine the total cost, we multiply the total pounds of turkey needed by the cost per pound: 30 pounds × $0.79/pound. This calculation gives us a total cost of $23.70 for the turkey.
Therefore, it will cost $23.70 to give turkey to 100 students.
The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers.
The volume of the square prism with a side of base decreasing at a rate of 7 km/min and height increasing at 10 km/min is decreasing at a rate of 344 cubic km/min.
Explanation:In this problem, we're dealing with rates of change and the geometric properties of a square prism. A square prism can be characterized by the side length of its base (we'll call this length 's') and its height (h). The volume (V) of a square prism is given by the equation V = s^2 * h. We're told that the side length s is decreasing at a rate of 7 kilometers per minute (ds/dt = -7 km/min) and the height h is increasing at a rate of 10 kilometers per minute (dh/dt = 10 km/min).
To find the rate at which the volume is changing with respect to time, we can take the derivative of the volume equation with respect to time. Thus, dV/dt = 2*s*ds/dt*h + s^2*dh/dt. Substituting the given values, when s = 4 km and h = 9 km, we find: dV/dt = 2*4*(-7)*9 + 4^2*10 = -504 + 160 = -344 cubic kilometers per minute. Therefore, at the given instant, the volume of the square prism is decreasing at a rate of 344 cubic kilometers per minute.
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The rate of change of the surface area of the prism at that instant is -204 square kilometers per minute. Correct Option is Option C.
To solve this problem, let's first start by identifying the formula for the surface area (SA) of a square prism, which is given as [tex]2s^2 + 4sh[/tex] , where s is the side of the base and h is the height.
Now, let's find the rate of change of the surface area with respect to time, dSA/dt. We'll use the chain rule to differentiate the surface area formula:
1. Differentiate the given surface area formula:
[tex]SA = 2s^2 + 4sh[/tex]
[tex]d(SA)/dt = d(2s^2)/dt + d(4sh)/dt[/tex]
2. Apply the chain rule:
d(SA)/dt = 2 * 2s * (ds/dt) + 4 * (s * (dh/dt) + h * (ds/dt))
3. Plug in the given values: s = 4 km, ds/dt = -7 km/min, h = 9 km, dh/dt = 10 km/min:
d(SA)/dt = 4 * 4 * (-7) + 4 * (4 * 10 + 9 * (-7))
d(SA)/dt = -112 + 4 * (40 - 63)
d(SA)/dt = -112 + 4 * (-23)
d(SA)/dt = -112 - 92
d(SA)/dt = -204
The rate of change of the surface area of the prism at that instant is -204 square kilometers per minute. Correct Option is Option C.
Complete Question:- The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers. What is the rate of change of the surface area of the prism at that instant (in square kilometers per minute)? Choose 1 answer: A 204 B 148 C -204 D -148 The surface area of a square prism with base side 8 and height h is [tex]2s^2+4sh.[/tex]
If you invest $500 at 3% compounded monthly for 2 years, how much interest you do earn? Show work.
Answer:
$30.88
Step-by-step explanation:
The account value is given by ...
A = P(1 +r/n)^(nt)
where P is the principal invested, r is the annual interest rate, t is the number of years, n is the number of times interest is compounded per year.
The amount of interest earned is the account value less the initial investment:
I = A - P = P(1 +r/n)^(nt) -P = P((1 +r/n)^(nt) -1)
Filling in the given values, we get ...
I = 500((1 +.03/12)^(12·2) -1) = 500(1.0025^24 -1) ≈ $30.88
The amount of interest earned is $30.88.
A construction worker leans his ladder against a building making a 60 degree angle with the ground. If the ladder is 20 feet long, how far away is the base or the ladder from the building? Round to the nearest tenth
Answer: the distance from the base of the ladder from the building is 10 feet.
Step-by-step explanation:
The ladder forms a right angle triangle with the wall of the building and the ground.
The length of the ladder represents the hypotenuse of the right angle triangle.
The distance, h of the base of the ladder from the building represents the adjacent side of the triangle.
To determine h, we would apply
the cosine trigonometric ratio.
Cos θ = adjacent side/hypotenuse. Therefore,
Cos 60 = h/20
h = 20Cos60 = 20 × 0.5
h = 10 feet
Final answer:
To solve for the distance from the base of the ladder to the building, the cosine function is used because the scenario forms a right-angled triangle with the ground and the wall. By the equation cos(60°) = Base / Hypotenuse, with the ladder being the hypotenuse, the base is calculated to be 10 feet from the building.
Explanation:
The question involves using trigonometry to determine the distance from the base of the ladder to the building. When a ladder makes a 60° angle with the ground and its length is given (20 feet), this forms a right-angled triangle with the ground and the building's wall. To find the distance from the base of the ladder to the building, we can use the cosine function, which relates the adjacent side to the hypotenuse in a right-angled triangle.
Using the equation:
cos(60°) = Base / Hypotenuse
Here, the hypotenuse is the length of the ladder which is 20 feet. Therefore:
cos(60°) = Base / 20 feet
cos(60°) equals 0.5 when calculating in degrees. Consequently:
0.5 = Base / 20 feet
Multiplying both sides by 20 gives:
Base = 20 feet * 0.5
Base = 10 feet
Thus, the base of the ladder is 10 feet from the building.
Bryce, a previous winner of the contest, made a trip of 360 miles in a 6.5 hours. At this same average rate of speed, how long will it take Bryce to travel an additional 300 miles so that he can judge the contest?
Answer:
5.42 hours
Step-by-step explanation:
Let x represent time taken to complete 300 miles.
We have been given that Bryce a previous winner of the contest, made a trip of 360 miles in a 6.5 hours.
We will use proportions to solve our given problem since rates are same for both distances.
[tex]\frac{\text{Distance covered}}{\text{Time taken}}=\frac{360\text{ Miles}}{6.5\text{ Hours}}[/tex]
Upon substituting our given values in above proportion, we will get:
[tex]\frac{300\text{ Miles}}{x}=\frac{360\text{ Miles}}{6.5\text{ Hours}}[/tex]
Cross multiply:
[tex]x\cdot 360\text{ Miles}=300\text{ Miles}\times6.5\text{ Hours}}[/tex]
[tex]x=\frac{300\text{ Miles}\times6.5\text{ Hours}}{ 360\text{ Miles}}[/tex]
[tex]x=\frac{300\times6.5\text{ Hours}}{360}[/tex]
[tex]x=5.41666\text{ Hours}\approx 5.42\text{ Hours}[/tex]
Therefore, it will take approximately 5.42 hours to travel additional 300 miles.
If cos(θ) = 6/8 and θ is in the IV quadrant, then fine:
(a) tan(θ)cot(θ)
(b) csc(θ)tan(θ)
(c) sin^2(θ) + cos^2(θ)
Answer:
a) 1
b) [tex]\frac{4}{3}[/tex]
c) = 1
Step-by-step explanation:
We are given the following in the question:
[tex]\cos \theta = \dfrac{6}{8}[/tex]
θ is in the IV quadrant.
[tex]\sin^2 \theta + \cos^2 \theta = 1\\\\\sin \theta = \sqrt{1-\dfrac{36}{64}} = -\dfrac{2\sqrt7}{8}\\\\\tan \theta = \dfrac{\sin \theta}{\cos \theta} = -\dfrac{2\sqrt7}{6}\\\\\csc \theta = \dfrac{1}{\sin \theta} = -\dfrac{8}{2\sqrt7}[/tex]
Evaluate the following:
a)
[tex]\tan \theta\times \cot \theta =\tan \theta\times\dfrac{1}{\tan \theta} = 1[/tex]
b)
[tex]\csc \theta\times \tan \theta\\\\= -\dfrac{8}{2\sqrt7}\times -\dfrac{2\sqrt7}{6} = \dfrac{4}{3}[/tex]
c)
[tex]\sin^2 \theta + \cos^2 \theta = 1\\\text{using the trignometric identity}[/tex]
Write an expression for the number or floors the building can have for a given building height. Tell what variable in your expression represents. (Each floor will be 12 feet tall)
Final answer:
To calculate the number of floors a building can have for a given height, use the formula 'Number of Floors = h / 12', where 'h' is the total building height in feet and each floor is 12 feet tall.
Explanation:
To write an expression for the number of floors a building can have for a given building height, where each floor is 12 feet tall, we use the following formula:
Number of Floors = Total Building Height (in feet) / Height of One Floor (in feet)
Let 'h' represent the total height of the building in feet. Then the expression becomes:
Number of Floors = h / 12
For example, if a building has a height of 384 feet, the number of floors can be calculated as 384 feet / 12 feet per floor = 32 floors.
A car with an initial cost of $23,000 is decreasing in value at a rate of 8% each year. Write the exponential decay function described in this situation. Then use your function to determine when the value of the car will be $15,000, to the nearest year.
Answer:
Step-by-step explanation:
We would apply the formula for exponential decay which is expressed as
A = P(1 - r/n)^ nt
Where
A represents the value after t years.
n represents the period for which the decrease in value is calculated
t represents the number of years.
P represents the value population.
r represents rate of decrease.
From the information given,
P = 23000
r = 8% = 8/100 = 0.08
n = 1
Therefore, the exponential decay function described in this situation is
A = 23000(1 - 0.08/n)1)^ 1 × t
A = 23000(0.92)^t
If A = 15000, then
15000 = 23000(0.92)^t
0.92^t = 15000/23000 = 0.6522
Taking log of both sides to base 10
Log 0.92^t = log 0.6522
tlog 0.92 = log 0.6522
- 0.036t = - 0.1856
t = - 0.1856/- 0.036
t = 5 years to the nearest year
The value of the car will be $15,000 after approximately 4 years.
To model the situation with an exponential decay function, we use the formula:
[tex]\[ V(t) = P \times (1 - r)^t \][/tex]
Given that the initial cost [tex]P[/tex] is $23,000 and the annual depreciation rate r is 8%, or 0.08 in decimal form, we can write the exponential decay function as:
[tex]\[ V(t) = 23000 \times (1 - 0.08)^t \][/tex]
[tex]\[ V(t) = 23000 \times (0.92)^t \][/tex]
To find out when the value of the car will be $15,000, we set [tex]V(t)[/tex] equal to $15,000 and solve for t.
[tex]\[ \frac{15000}{23000} = (0.92)^t \][/tex]
[tex]\[ 0.652173913043478 = (0.92)^t \][/tex]
Using a calculator, we find:
[tex]\[ t \approx \frac{\ln(0.652173913043478)}{\ln(0.92)} \approx 4.037 \][/tex]
Since we are looking for the time in whole years, we round to the nearest year:
[tex]\[ t \approx 4 \][/tex]
Therefore, the value of the car will be $15,000 after approximately 4 years.
Suppose Rick has 40 ft of fencing with which he can build a rectangular garden.Letxrepresent the length of the garden and letyrepresent the width.(a) Write and inequality representing the fact that the total perimeter of thegarden is at most 40 ft.(b) Sketch part of the solution set for this inequality that represents all possiblevalues for the length and with of the garden. (Hint:Note that both the lengthand the width must be positive.)
Answer:
20 >= x + y
Step-by-step explanation:
Given:
- The length of garden = x
- The width of the garden = y
- Total fence available = 40 ft
- Rectangular garden
Find:
(a) Write and inequality representing the fact that the total perimeter of thegarden is at most 40 ft.
(b) Sketch part of the solution set for this inequality that represents all possiblevalues for the length and with of the garden.
Solution:
- The perimeter of the rectangular garden is P at most 40 ft:
P >= 2*x + 2*y
40 >= 2*x + 2*y
20 >= x + y
- The sketch of the graph will be all points in the shaded region denoted by the inequality as follows:
y =< 20 - x
- See the triangular shaded region.
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
Since Rick has 40ft of fencing
Then, the perimeter cannot be more than 40ft if he decided to lay the block on a single layer and not on each other
Then if the length is x
And the breadth is y
Perimeter of a rectangle is 2(l+b)
Therefore,
Perimeter is less than or equal to 40ft
2(l+b) ≤ 40
b. 2(l+b)≤ 40
Then divide both side by 2
l+b≤ 20.
Then l ≤ 20-b
Also, b ≤ 20-l
Check attachment for graph
Mary purchased an annuity that pays her $500 per month for the rest of her life. She paid $70,000 for the annuity. Based on IRS annuity tables, Mary's life expectancy is 16 years. How much of the first $500 payment will Mary include in her gross income (round to two decimals)?
Answer: $135.42
Step-by-step explanation:
Based on IRS tables, Mary is expected to receive 192 (16 years x 12 months) annuity payments. Her investment in the annuity is $70,000 and her return of capital for each annuity payment is $70,000/192 = $364.58. The return of capital portion of each annuity payment is not taxable (not included in gross income). Mary must include the excess received ($500.00 – 364.58) of $135.42 in her gross income.
To calculate how much of the first $500 payment Mary will include in her gross income, we need to determine the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary will include $36.45 of the first $500 payment in her gross income.
Explanation:To determine how much of the first $500 payment Mary will include in her gross income, we need to calculate the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary paid $70,000 for the annuity and her life expectancy is 16 years, so the exclusion ratio is $70,000 / ($500/month * 12 months/year * 16 years) = 0.0729. To determine the amount she will include in her gross income, we multiply the monthly payment by the exclusion ratio: $500 * 0.0729 = $36.45. Therefore, Mary will include $36.45 of the first $500 payment in her gross income.
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A rectangle has width w inches and height h, where the width is twice the height. Both w and h are functions of time,t, measured in seconds. If A represents the area of the rectangle, what is the rate of change of A with respect to t at the instant where the width is 4 inches and the height is increasing at the rate of 2 inches per second?
Answer:
dA/dt = 16 square inches per second
Step-by-step explanation:
Width of rectangle is w
Height of rectangle is h
width = twice height
w = 2h
Area = wh = (2h)*h
A = 2h^2
Differentiating the equation with respect to time
dA/dt = 2+2h dh/dt
dA/dt = 4h dh/dt
According to the given situation when width is 4 inches
h = w/2
h = 2
Rate of change of A is
dA/dt = wh dh/dt
dA/dt = 4(2)(2)
dA/dt = 16 square inches per second
The rate of change of the area of the rectangle with respect to time at the specified instant is 16 square inches per second.
Explanation:The question asks for the rate of change of the area of a rectangle with respect to time, where the rectangle's width is twice its height, both width and height are functions of time and the rate at which height is increasing is given.
The area A of the rectangle is given by the formula A = w*h. Substituting w = 2h into the formula gives A = 2h². The rate of change of the area with respect to time (dA/dt) can be found by differentiating A with respect to time. By the chain rule this gives dA/dt = 2*2h(dh/dt) = 4h*dh/dt.
At the given instant, where the width is 4 inches (so the height is 2 inches) and the rate of change of the height is 2 inches per second, we have dA/dt = 4*2*2 = 16 square inches per second. Therefore, the rate of change of the area of the rectangle with respect to time at that instant is 16 square inches per second.
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Aaron and Blake both wrote down equation. Their equation has the same slope, But Aaron's y-intercept was negative and Blake's y-intercept was positive. How many solutions for their system of equations have? Explain your reasoning
Answer:
Parallel lines do not intersect therefore the system of equations cannot have any solution. Therefore there no solutions to the given system of equations.
Step-by-step explanation:
Aaron and Blake both wrote down equation. Their equation has the same slope, But Aaron's y-intercept was negative and Blake's y-intercept was positive.
Therefore both equations will represent lines that are parallel.
Parallel lines do not intersect therefore the system of equations cannot have any solution.
PLEASE HELP ASAP!!! I NEED CORRECT ANSWERS ONLY PLEASE!!!
Find m∠D.
Write your answer as an integer or as a decimal rounded to the nearest tenth.
m∠D = °
Answer:
27° (as an integer)
Step-by-step explanation:
From the figure given;
Triangle BCD is a right triangle;We are given the two shorter sides;DC = 8 units
BC = 4 units
We are required to determine m∠D
We need to use the appropriate trigonometric ratio;
In this case, DC is adjacent and BC is opposite to m∠D
Therefore, the appropriate trigonometric ratio is tangent;
That is;
Tan m∠D = BC ÷ DC
= 4 ÷ 8
= 0.5
Thus;
m∠D = Tan^-1 (0.5)
= 26.57
= 27 (as an integer)
Thus, m∠D is 27°