Answer:
x = 137
Step-by-step explanation:
All quadrilaterals have a sum in angles of 360
We can see the parallel angles are the same
One angle is 43 degrees
The angle to the left (if your looking at the picture right side up) is the same angle
subtract is by 360 because the total degrees is 360
360 - 43 = 317
317 - 43 = 274
Then divide 274 by 2 because the two angles that are remaining are the same degrees
274 divided by 2 = 137
x = 137
a marksman has a probability of 0.268 for hitting a certain long-range target. what is the probability that it takes 5 shots for the marksman to hit the long-range target? (round your answer to three decimal places, if necessary.)
The probability of the marksman hitting the long-range target in 5 shots is 0.1429.
The geometric distribution is a Bernoulli experiment that predicts the likelihood of the first success after a certain number of failures. Negative-binomial and binomial distributions are two different forms of Bernoulli trials.
P = 0.268
x = 5
Use the geometric probability mass function below to calculate the probability:
P(X = x) = \((1 - P)^{(x-1)}\)× P
P(X = x) = \((1 - 0.268)^{(3-1)}\)× 0.268
P(X = x) = 0.1429
The probability that it takes 5 shots for the marksman to hit the long-range target is 0.1429.
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what is 0.123 with the 23 repeating
Answer:
The answer is 61/495 simplified.
how do you simplifty the square root of 48 times the square root of 3?
Answer:
\( \sqrt{48} \times \sqrt{3} \\ \sqrt{144} \\ 12\)
In a company's first year in operation, it made an annual profit of $112,000. The profit of the company increased at a constant 18% per year each year. How much total profit would the company make over the course of its first 26 years of operation, to the nearest whole number?
If the company made a profit of $112000, i first year operation, then the total profit made by the company in 26 years of operation is $8170286.
In order to find the profit after 26 years, we use the formula for "future-value" of investment with compound interest;
⇒ FV = PV × (1 + r)ⁿ,
where FV is = future value, PV is = present value, r is = interest rate per period, and n = time (in years),
In this case, the "present-value" is $112,000, the "interest-rate" per period is 18%, and time is 26 years.
We have to find the total profit, which is = "future-value" - "present-value",
⇒ Total profit = FV - PV,
Substituting the value,
We get,
⇒ FV = $112000 × (1 + 0.18)²⁶,
⇒ FV = $8282285.79 ≈ $8282286,
So, Total profit = $8282286 - $112000,
Total profit = $8170286,
Therefore, the company would make a total-profit of $8170286.
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3) What happens when you add a pair of opposites together?
The sum of any integer and its opposite is equal to zero
Hope this helps :)
Haley bought 5 boxes of pencils. She is going to give out the pencils to 10 of her friends. She wants to give each friend the same number of pencils. What does Haley need to know to figure out how many pencils she can give each friend?
We are given the quantities :
+) The number of boxes of pencils
+) The number of her friends
She cannot hand out pencils in boxes to her friends, as she has less boxes of pencils than the number of her friends (assuming they cannot share boxes of pencils.)
Therefore, the quantity we need to know is : The number of pencils in each box of pencils.
Jeffrey wants to make jam. He needs a combination of raspberries and blackberries totaling six pounds. Blueberries cost $1.00 per pound, and raspberries cost $2.00 per pound. If Jeffrey can afford $11.60, how many pounds of each berry should he buy?
The amount for each berry are 5.60 and 1.60 respectively.
What is equation?An equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =
Given that, Jeffrey wants to make jam. He needs a combination of raspberries and blackberries totaling six pounds. Blueberries cost $1.00 per pound, and raspberries cost $2.00 per pound. If Jeffrey can afford $11.60,
We will set a system of linear equations, to solve this,
Establishing the equation,
Let the amount of raspberries be x and that of blueberries be y,
x + y = 6
y = 6-x...(i)
2x + y = 11.60
y = 11.60 - 2x...(ii)
Equating eq(i) and eq(ii)
6-x = 11.60 -2x
x = 5.60
Put x = 5.60 in eq(i)
y = 6-5.60
y = 1.60
Hence, the amount for each berry are 5.60 and 1.60 respectively.
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A train is traveling at a constant speed and goes a distance of 7 1/2 kilometers in 6 minutes the train will continue at this speed the train will travel a distance of d kilometers in m minutes write an equation that represents the relationship between d and m
Answer:
392939939393939399393939393
Step-by-step explanation:
An equation that represents the relationship between d and m can be written as d = 1.25 m.
What is Speed?Speed of an object is the rate of change of position of an object in any direction.
Speed is a scalar quantity because it does not have any direction, it only has magnitude. Speed with direction is called velocity.
The formula for calculating speed is,
Speed = Distance / Time
Given distance the train travelled = 7 \(\frac{1}{2}\) kilometers
= 15/2 kilometers
= 7.5 kilometers
Time taken to travel the distance = 6 minutes
Speed = 7.5 / 6
= 1.25 kilometers / minute
Given that speed of the train is constant.
Hence for any distance 'd' kilometers in 'm' minutes,
d / m = 1.25
d = 1.25m
Hence d and m can be related by the equation d = 1.25m.
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Reflect the given triangle over the line y=-x. 3 6 3 -3 3 3 [?] [ -3 [ ] [ 3
The points of the vertices of the triangle after reflection on the line y = -x is given by A'(3,-3) , B'(-3,-6) and B'(-3,-6) .
We know that when a point is reflected around a line on the coordinate axes then the coordinates reflect following some rules.
Now when the point (a,b) is reflected around the line y = -x then ,the coordinates of the image becomes (-b,-a)
Therefore :
A(3,-3)→A'(3,-3 , does not change as the point lies on the line y=-x)
B(6,3)→B'(-3,-6)
C(3,3)→B'(-3,-6)
In a transition known as reflection over a line k, each point of the original figure (pre-image) must have an image that is the same distance from the line of reflection as the first comment but is on the other side of the line. Remember that a flip is a reflection. The size of the figure does not alter when it is reflected.
A figure created by a line reflection that is consistent with the original figure is called an isometry (a transformation that preserves length). Because naming the figure in a reflection implies changing the order of the letters, a reflection is more accurately referred to as a non-direct or opposing isometry .
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Answer: 3 -3 -3
-3 -6 -3
Step-by-step explanation:
We know that there is likely to be interaction between these two factors. ... optimum levels of two fertilizer ingredients, nitrogen (N) and phosphorus (P).
The statement suggests that there is likely to be an interaction between nitrogen (N) and phosphorus (P), which are two fertilizer ingredients used to achieve optimum levels in plant growth. However, without further context or specific details, it is uncertain to determine the nature and extent of this interaction.
In agriculture and plant nutrition, both nitrogen and phosphorus are essential elements for plant growth and development. They play crucial roles in various physiological processes and are often supplied through fertilizers to optimize crop yields. The interaction between nitrogen and phosphorus can have significant implications for plant growth and nutrient utilization.
The ratio of nitrogen to phosphorus in the soil can affect nutrient uptake, plant growth, and overall productivity. Different plant species and soil conditions may exhibit varying responses to the interaction between these two fertilizer ingredients. Factors such as soil type, pH, organic matter content, and crop-specific nutrient requirements further contribute to the complexity of this interaction. To fully understand and assess the interaction between nitrogen and phosphorus, detailed research and experimentation specific to the particular context are necessary.
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HELP PLEASE FAST
Solve for x
Answer:
\(x^o+2x^o=180^o [Sum~of~angles~of~straight~line~is~180^o]\\or, 3x^o=180^o\\or, x = 60^o\)
Step-by-step explanation:
x + 70° = 130° {exterior angle of a triangle is equal to the sum of two opposite interior angles}
x = 130° - 70°
x = 60°
Now for 2x = 2 * 60° = 120°
Yen Luong owns a used pickup truck that she wants to sell. A used-vehicle guide shows that its average retail value is $17,600. She adds $1,700 for 4-wheel drive, $475 for an entertainment system, $800 for a special trim package, and $225 for power locks. She aiso adds $125 for a sliding rear window, $325 for a towing package, ers0 for power windows, and $3,125 for a diesel engine. She deducts 3615 for having a manual transmission. She adds $450 for having less Tham the expected mileage. What is the average retail price of Yen's
vehicle?
The answer we have states that the typical retail cost of Yen's car is $21,210.
Is the retail cost the actual cost?The price that the automaker, or manufacturer, advises that the seller ask for the vehicle is known as the Mfg Suggested Retail Price (MSRP). It is not necessary for it to be the price you really pay. Many buyers haggle to get the automobile for less than the MSRP.
We must add the numbers she added and minus the sums she reduced to obtain the vehicle's average retail price.
$17,600 + $1,700 + $475 + $800 + $225 + $125 + $325 + $3,125 + $450 = $24,825
And let's subtract $3,615 as from manual transmission's cost, and brings us back amount approximately $21,210.
Hence, Yen's car has an average retail price of $21,210.
Therefore, the average retail price of Yen's vehicle is $21,210 .
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Is the new distance less than 10, equal to 10, or greater than 10 times the original distance?
The correct option is (1) less than 10 times the original distance.The original distance was 30 cm and the new distance is 94.86 cm which is less than 10 times the original.
What is Coulomb's force law?The force of attraction and repulsion among two charged bodies are directly proportional to a product of their charges but inversely proportional to a square of a distance between them, according to Coulomb's law.
It works along the line that connects the two charges that are called point charges.
The formula for Coulomb's law is-
\(f_{}=\frac{q_{1} q_{2}}{4 \pi \epsilon d_{}^{2}}\)
where, q₁ and q₂ are the charges;
d is the distance between them.
Now, according to the question;
Two charges that were initially separated by a particular distance are pushed closer together until force between the two is reduced by a factor of ten.
Coulomb's law of force
\(f_{1}=\frac{q_{1} q_{2}}{4 \pi \epsilon d_{1}{ }^{2}}\)
d₁ is the initial distance.
So when charges are separated further, a new force is f₂, which is represented as
\(f_{2}=\frac{q_{1} q_{2}}{4 \pi \epsilon d_{2}^{2}}\)
d₂ is the moved distance
As a result, the given force is decreased by ten times its original value.
Thus, \(f_{2}=\frac{f_{1}}{10}\)
use the expression in both forces
\(\begin{gathered} \frac{f_{1}}{10}=\frac{q_{1} q_{2}}{4 \pi \epsilon d_{2}{ }^{2}} \\\frac{q_{1} q_{2}}{4 \times 10 \pi \epsilon d_{1}{ }^{2}}=\frac{q_{1} q_{2}}{4 \pi \epsilon d_{2}{ }^{2}} \\ \frac{1}{10 \times d_{1}{ }^{2}}=\frac{1}{d_{2}{ }^{2}} \\ d_{2}=\sqrt{10} d_{1}\end{gathered}\)
Now, calculate the moved distance d₂,
\(\begin{aligned}& d_{2}=\sqrt{10} d_{1} \\ & d_{2}=\sqrt{10} \times 30 \\& d_{2}=3.162 \times 30 \\ & d_{2}=94.868 \mathrm{~cm}\end{aligned}\)
The new distance is 94.868 cm.
Therefore, the new obtained distance is less than the 10 times the original distance.
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The complete question is-
Two charges originally separated by a certain distance are moved farther apart until the force between them has decreased by a factor of 10. Is the new distance
(1) less than 10,
(2) equal to 10, or
(3) greater than 10;
times the original distance? If the original distance was 30 cm.
Let f(x) be a function defined for all positive real numbers satisfying the conditions f(x)>0 for all x>0 and f(x−y)=√f(xy)+1 for all x>y>0. Determine f(2009).
After evaluation, the resultant value of the function f(2009) is 1.9021.
What are functions?A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X.
The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
Any input for x results in a single output for y.
Considering that x is the input value, we would state that y is a function of x.
So, evaluate f(2009) as follows:
Given that f(x) is a function with all positive real numbers satisfying the requirement that f(x) > 0 and for all x > 0 and also:
\(\begin{aligned}&f(x-y)=\sqrt{f(x y)+1} \\&x > 0, y > 0 \\&\text { for, } x=1, \text { and } y=\frac{1}{2} \\&f\left(1-\frac{1}{2}\right)=\sqrt{f\left(1 \times \frac{1}{2}\right)+1} \\&f\left(\frac{1}{2}\right)^2=f\left(\frac{1}{2}\right)+1 \\&f\left(\frac{1}{2}\right)=\frac{1+\sqrt{5}}{2}\end{aligned}\)
Now, consider: x = 2009 and y = 0
\(\begin{aligned}&f(2009-0)=\sqrt{f(2009 * 0)+1} \\&f(2009)=\sqrt{f(0)+1} \\&f(2009)=\sqrt{f\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\right)+1} \\&f(2009)=\sqrt{\sqrt{f\left(\frac{1}{\sqrt{2}} * \frac{1}{\sqrt{2}}\right)+1}+1} \\&f(2009)=\sqrt{\sqrt{f\left(\frac{1}{2}\right)+1}+1} \\&f(2009)=\sqrt{\sqrt{\frac{1+\sqrt{5}}{2}+1}+1} \\&f(2009)=\sqrt{\sqrt{\frac{3+\sqrt{5}}{2}}+1} \\&f(2009)=\sqrt{\sqrt{5.2362}+1} \\&=\sqrt{3.6180} \\&=1.9021\end{aligned}\)
Therefore, after evaluation, the resultant value of the function f(2009) is 1.9021.
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Can someone please help me ASAP
Answer:
D
Step-by-step explanation:
They should both level out at the same height considering the volume of hydrochloric acid used.
Answer:
D
Step-by-step explanation:
Math is not my favorite subject XD so help would be helpful
Answer:
g(x) = g = 3(x)
g(-10) = -30
Step-by-step explanation:
as we can see in the pattern, x times 3 is equal to g(x) -->
so we can conclude that g=3(x) which means g is 3 then multiply by x
finding g(-10) = 3(-10) => -30 and thus, we have our answers
Hope this helps ad sorry if it's a little not clear!
Use the normal approximation to find the indicated probability. The sample size is n, the population proportion of successes is p, and X is the number of successes in the sample.
n = 83, p = 0.47: P(X ≥ 34)
Probability of getting at least 34 successes in a sample of 83 with a population proportion of 0.47 using the normal approximation is approximately 0.1056 or 10.56%.
To use the normal approximation, we need to check if the sample size and the population proportion satisfy the conditions of the Central Limit Theorem. In this case, since n*p = 39.01 and n*(1-p) = 43.99 are both greater than 10, we can assume that the sampling distribution of X is approximately normal.
To find P(X ≥ 34), we can use the normal distribution with mean\(µ = n*p = 39.01\) and standard deviation σ = sqrt(n*p*(1-p)) = 4.01. Then, we need to standardize the value of X using the formula z = (X - µ) / σ, which gives:
z = (34 - 39.01) / 4.01 = -1.25
Using a standard normal table or calculator, we can find the probability of z being less than -1.25, which is equivalent to the probability of X being greater than or equal to 34, as:
P(Z < -1.25) = 0.1056
Therefore, the probability of getting at least 34 successes in a sample of 83 with a population proportion of 0.47 using the normal approximation is approximately 0.1056 or 10.56%.
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how many terms are there in the series 7+21+15+19+.....+79?
anybody can answer these .
Answer:
7+21+15+19+.....+79
Here 21 be replaced by 11.
Now., common diffrence btw terms , d = 11-7 = 4
first term , a = 7
Last term , l = 79
number of terms = ( a + l ) ÷ d
=( 7 + 79 ) ÷ 4 = 86 ÷ 4 = 21.5 { It cannot be in decimal }
There's some mistake in the terms you provided please check your question again...
I've been very confused for hours now on when to put an imfity sign when doing quadratic inequalities pls help
Answer:
The symbol (∞) is read as infinity. and indicates that the set is unbounded to the right on a number line. Interval notation requires a parenthesis to enclose infinity. The square bracket indicates the boundary is included in the solution.
Step-by-step explanation:
7.34 the gpas of all students enrolled at a large university have an approximate normal distribution with a mean of 3.02 and a standard deviation of .29. find the probability that the mean gpa of a random sample of 20 students selected from this university is
The probability that the mean GPA of a random sample of 20 students selected from this university is 3.10 or higher is 10.93%
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution:Problems of normally distributed samples are solved using the z-score methods.
We know that, in a set with mean and standard deviation , the z-score of a measure X is given by:
Z = X - μ/σ
After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem:The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .
For a Skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
μ = 3.02 , σ = 0.29 , n = 20 and s = 0.29/√ 20 = 0.0648
Find the probability that the mean GPA of a random sample of 20 students selected from this university is 3.10 or higher.
This is 1 subtracted by the p- value of Z when X = 3.10. So
By the Central Limit Theorem:
Z = X - μ / s
⇒ Z = 3.1 - 3.02/ 0.0648
⇒ Z = 1.23
The Z = 1.23 has a P- value of 0.8907
∴ 1 - 0.8907 = 0.1093
Now, for getting the percentage,
0.1093 × 100% = 10.93%
10.93% probability that the mean GPA of a random sample of 20 students selected from this university is 3.10 or higher.
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mini earned $160 during the summer doing chores. she bought 3 dresses worth $12 each using her chore money. how much money was left after she bought the dresses.
a. $112
b. $124
c. $136
d. $148
Answer:
124
Step-by-step explanation:
Answer:
The answer is $124
Step-by-step explanation:
Look here ez dubs
Solve for x. Round to the nearest tenth.
Using trigonometric function the value of x is 51.3°.
What is trigonometric function?
The functions of an angle in a triangle are known as trigonometric functions, commonly referred to as circular functions. In other words, these trig functions provide the relationship between a triangle's angles and sides. There are five fundamental trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.
Here Let us take the given right triangle as ABC.
∠A = x , ∠B = 90° and AB = 25 , AC= 40
Now using Cosine ratio then
Cos A = \(\frac{adjacent}{hypotenuse}\)
=> cos x = \(\frac{25}{40}\)
=> cos x = \(\frac{5}{8}\)
=> x = \(cos^{-1}\frac{5}{8}\)
=> x = 51.3°
Hence the value of x is 51.3°.
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Json has 4 gallons of hand sanitizer that contains 25% alcohol. He want sto make a hand santizer that contains 50% alchahol. How many gallons of pure alcohol should he add?
Answer:
2
Step-by-step explanation:
1+x/4+x = 1/2
4+x = 2 + 2x
x = 2
There are 170 deer on a reservation. The deer population is increasing at a rate of 30% per year
A function \(P(t) = 170.(1.30)^t\) that gives the deer population P(t) on the reservation t years from now
We were told there were 170 stags on reservation. The number of deer is increasing at a rate of 30% per year.
We could see the deer population grow exponentially since each year there will be 30% more than last year.
Since we know that an exponential growth function is in form:
\(f(x) = a*(1+r)^x\)
where a= initial value, r= growth rate in decimal form.
It is given that a= 170 and r= 30%.
Let us convert our given growth rate in decimal form.
\(30 percent = \frac{30}{100} = 0.30\)
Upon substituting our given values in exponential function form we will get,
\(P(t) = 170.(1+0.30)^t\)
⇒ \(P(t)= 170.(1.30)^t\)
Therefore, the function \(P(t) = 170.(1.30)^t\) will give the deer population P(t) on the reservation t years from now.
Complete Question:
There are 170 deer on a reservation. The deer population is increasing at a rate of 30% per year. Write a function that gives the deer population P(t) on the reservation t years from now.
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The following question is about the rational function r(x) = (x + 1)(x - 3)/(x + 3)(x - 7). The function r has y-intercept __________. The following question is about the rational function r(x) = (x + 1)(x - 3)/(x + 3)(x - 7) The function r has vertical asymptotes x = ______ (smaller value) and x = __________ (larger value).
The function r(x) = (x + 1)(x - 3)/(x + 3)(x - 7) has a y-intercept of -2/3.
The rational function r(x) = (x + 1)(x - 3)/(x + 3)(x - 7) has a y-intercept when x = 0.
Plugging in x = 0, we get r(0) = (0 + 1)(0 - 3)/(0 + 3)(0 - 7)
Which simplifies to r(0) = (-1)(-3)/(-7)(3), resulting in r(0) = 1/7.
So, the y-intercept is (0, 1/7).
The function also has vertical asymptotes at x = -3 (smaller value) and x = 7 (larger value).
The function r has vertical asymptotes at the values of x where the denominator is equal to zero.
This occurs when (x + 3) = 0 and (x - 7) = 0.
Solving these equations, we find the vertical asymptotes at x = -3 (smaller value) and x = 7 (larger value).
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To find the y-intercept of r(x), we plug in x = 0: r(0) = (0 + 1)(0 - 3)/(0 + 3)(0 - 7) = -3/21 = -1/7. Therefore, the function r has a y-intercept of -1/7.
To find the vertical asymptotes of r(x), we set the denominators of the fractions equal to zero:
x + 3 = 0 and x - 7 = 0
Solving for x, we get:
x = -3 and x = 7
Therefore, the function r has vertical asymptotes at x = -3 (smaller value) and x = 7 (larger value).
To find the y-intercept of the rational function r(x) = (x + 1)(x - 3)/(x + 3)(x - 7), we need to set x = 0 and solve for r(0):
r(0) = (0 + 1)(0 - 3)/(0 + 3)(0 - 7) = (1)(-3)/(3)(-7) = 3/7
So, the y-intercept is at (0, 3/7).
Now, to find the vertical asymptotes, we look at the denominator of the rational function, which is (x + 3)(x - 7). The vertical asymptotes occur when the denominator equals 0. We set each factor equal to 0 and solve for x:
x + 3 = 0 → x = -3 (smaller value)
x - 7 = 0 → x = 7 (larger value)
So, the function r has vertical asymptotes at x = -3 and x = 7.
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What is the value of x in the equation below?
6(x-3) = 72
12
15
20
24
Answer:B
Step-by-step explanation:
Write a congruence statement for the pair of triangles. Name the postulate it theorem that justifies your statement.
Answer:
im so sorry
Step-by-step explanation:
I just came to look at it and I don't think I can solve it I'm so sorry I'm really sorry I hope you get it right
which statement is correct? int k = 15 / 4; byte b = 100 * 1000; float f = 3.5 * 4; int k = (double) 4 * (int) 4.5; none of them
None of the given statements is correct
1. int k = 15 / 4; - This statement performs integer division, resulting in a quotient of 3. Therefore, k would be assigned the value 3.
2. byte b = 100 * 1000; - This statement performs integer multiplication, resulting in the product of 100,000. However, the value 100,000 is outside the range of a byte (-128 to 127), so it cannot be assigned to a byte variable.
3. float f = 3.5 * 4; - This statement performs floating-point multiplication, resulting in the product of 14.0. This value can be assigned to a float variable, so the statement is valid.
4. int k = (double) 4 * (int) 4.5; - This statement attempts to cast 4 to a double and 4.5 to an int. However, the cast of 4.5 to an int truncates the decimal portion, resulting in a value of 4. Then, the multiplication is performed, resulting in a product of 16. However, the assignment to an int variable is not valid because it violates the syntax rules of declaring the same variable twice in the same scope.
None of the given statements is correct. Statement 2 attempts to assign a value outside the range of a byte, and statement 4 violates the syntax rules of variable declaration.
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triangle has an area of 230.86 square inches. The height of the triangle is 23.8 inches. What is the length of the base of the triangle?
19.4 inches
Step-by-step explanation:
Area of the traingle = 230.86 sq. inches
Height of the triangle = 23.8 sq. inches
We have,
Area of the triangle = 1/2bh
where,
b = base of the traingle
h = height of the triangle
Now, substituting the given info we get,
230.86 = 1/2b (23.8)
230.86 × 2/23.8 = b
b = 19.4
So, the base of the triangle is 19.4 inches
please help 20 points
Answer:
Option C is correct
Step-by-step explanation:
\(a. - 3, - 12 \\ = 36\)
\(b. - 6, - 6 \\ = 36\)
\(c. \: \: 9, - 4 \\ = - 36\)
\(d. \: \: 2,18 \\ = 36\)
Hope it is helpful...