You have learned about special right triangles. Using that knowledge, you decide to take a square cake with 10-inch sides and divide it diagonally to form two special right triangles. Find and explain how you know the exact (no decimals) measure of the hypotenuse.
Now find the exact diagonal measure of cakes that are an 8-inch square, a 12-inch square, and a 15-inch square.
Did you discover a shortcut that works with any square cake? Explain.
Please show your work
Answers
thus, using the 45-45-90 rule, for a square with sides if 8-inch, h = 8√(2), for 12-inch sides, h = 12√(2), for 15-inch sides, h = 15√(2), and so on.
The hypotenuse of a right triangle formed by dividing a square cake diagonally can be found using the Pythagorean Theorem, with the shortcut that the hypotenuse is the side length of the square multiplied by √2. Examples include a hypotenuse of 10√2 inches for a 10-inch square, 8√2 inches for an 8-inch square, 12√2 inches for a 12-inch square, and 15√2 inches for a 15-inch square.
Finding the Hypotenuse of Right Triangles in Squares
To find the length of the hypotenuse in a right triangle that is formed by dividing a square cake diagonally, we use the Pythagorean Theorem which states that for a right triangle with sides a and b and hypotenuse c, the relation is c = √(a ² + b ²). Since the sides of the square are equal in length, for a square with 10-inch sides, the hypotenuse will be √(10 inches ² + 10 inches ²) = √(100 + 100) = √200 inches = 10√2 inches.
For the other square cakes:
8-inch square: Hypotenuse = √(64 + 64) = 8√2 inches
12-inch square: Hypotenuse = √(144 + 144) = 12√2 inches
15-inch square: Hypotenuse = √(225 + 225) = 15√2 inches
The shortcut discovered is that the length of the hypotenuse is always the side length of the square multiplied by √2, because each side of the square forms a right triangle with equal sides when the square is divided diagonally.
Related Questions
Find the value of 4 + (27 - 12·2) ÷ 2.
Answer's are 3.5,5.5 and 19
Which one is it
Answers
The value of 4 + (27 - 12·2) ÷ 2 would be 5.5.
What is simplification of an expression?Usually, simplification involves proceeding with the pending operations in the expression.
Like, 5 + 2 is an expression whose simplified form can be obtained by doing the pending addition, which results in 7 as its simplified form.
A Simplification usually involves making the expression simple and easy to use later.
WE follow the rule PEMDAS parenthesis, exponent, multipy, divide, add, substart.
4 + (27 - 12·2) ÷ 2
4 + (27 - 24 ) ÷ 2
4 + (3) ÷ 2
4+3/2, do 3/2=1.5
then add it
4+1.5 = 5.5.
Therefore, the correct answer is 5.5
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Most teachers enjoy teaching students. Most teachers are college graduates. Assuming these two statements are true, do some college graduates enjoy teaching students?
Answers
Answer:
yes some some college graduates enjoy teaching students.
Step-by-step explanation:
This question is based on reasoning.
Most teachers enjoy teaching students.
Most teachers are college graduates.
A syllogism is a form of logical reasoning formed by the joining of two or more premises to arrive at a conclusion.
So, here we can say that yes some some college graduates enjoy teaching students.
What type of graph would have the title, "Monthly Company Profit for the Year?"
bar graph
line plot
line graph
pictograph
Answers
Answer:
line graph
Step-by-step explanation:
On a resent business trip it took 7 hours 30 minutes to drive 375 miles how many miles per hour was the average
Answers
Answer:
50 miles / hours.
Step-by-step explanation:
Speed may be defined as the distance covered by the object per unit time.
Speed = [tex]\frac{distance}{time}[/tex]
the distance of the business trip = 375 miles.
the time taken by the business trip = 7 hours 30 minutes = 7.5 hrs.
Speed = [tex]\frac{375}{7.5}[/tex]
Speed = 50 miles / hours.
Here, the average of miler per hour is 50 miles / hours.
The formula for finding the circumference of a circle is C=2πrC=2πr . Rearrange the formula to solve for the radius (r).
Answers
Divide by 2π:
r = C/2π
Valerie predicts she will be able to hold her breath for 31 second while underwater. If she actually held her breath for 34 seconds, what was Valerie's percent error? Round your answer to the nearest tenth of a percent.
Answers
(31 - 34)/34 * 100 = 8.8%
If she actually held her breath for 34 seconds. Then the percentage error will be 8.82%.
What is the percentage error?The amount of error is expressed as if it is a part of the total which is a hundred. The ratio can be expressed as a fraction of 100. The word percent means per 100. It is represented by the symbol ‘%’.
Valerie predicts she will be able to hold her breath for 31 seconds while underwater.
If she actually held her breath for 34 seconds.
Then the percentage error will be
[tex]\rm error \ \% = \dfrac{34-31}{34} \times 100\\\\\\error \ \% = \dfrac{3}{34} \times 100\\\\\\error \ \% = 8.82 \%[/tex]
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Find the volume of the wedge-shaped region contained in the cylinder x2 + y2 = 49, bounded above by the plane z = x and below by the xy-plane.
Answers
Converting to cylindrical coordinates, the integral is equivalent to
[tex]\displaystyle\iiint_R\mathrm dV=\int_{\theta=\pi/2}^{\theta=3\pi/2}\int_{r=0}^{r=7}\int_{z=r\cos\theta}^{z=0}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_{\theta=\pi/2}^{\theta=3\pi/2}\int_{r=0}^{r=7}-r^2\cos\theta\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=-\displaystyle\left(\int_{\theta=\pi/2}^{3\pi/2}\cos\theta\,\mathrm d\theta\right)\left(\int_{r=0}^{r=7}r^2\,\mathrm dr\right)[/tex]
[tex]=\dfrac{2\times7^3}3=\dfrac{686}3[/tex]
We have that
Volume of the wedge-shaped region
[tex]W=228.65[/tex]
Attached below is a picture of the wedge-shaped region contained in the cylinder
Bearing in mind this shape, we have that the coordinates identifying the cylinder are as follows
Co-ordinates
[tex]0 \leq r \leq 7,\frac{\pi}{2} \leq 0 \leq \frac{\pi}{2},0 \leq z \leq x[/tex]
Where
[tex]0 \leq z \leq x =rcos \phi[/tex]
Generally the equation for the volume of the wedge-shaped region W is mathematically given as
[tex]\int \int \int_W 1dv=\int^{7}_{0} \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \int^{rcos \phi}_{0}[/tex]
[tex]\int \int \int_W 1dv=\int^{7}_{0} r (\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}rcos\phi dv))dr[/tex]
[tex]\int \int \int_W 1dv=(\frac{(7)^3}{3}+\frac{(0)^3}{3})(sin {\pi/2}+sin{-\pi/2})[/tex]
[tex]\int \int \int_W 1dv=228.67[/tex]
Therefore
Volume of the wedge-shaped region
[tex]W=228.65[/tex]
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Jason deposits $5,000 in a bank account that will pay him 4% simple interest annually. If Jason deposits no more than the initial 5,000 , how much money will be in the account at the end of 5 years ?
Answers
Here I=$5000, r=(4/100)=0.04 so
F=$5000(1+0.04t) so after 5 years...
F=$5000(1+0.2)
F=$6000
A ratio is a rate. Sometimes always or never true
Answers
The statement given above, "A ratio is a rate" is only sometimes true. This is only true when the unit of measurement of the denominator is in terms of time (seconds, minutes, hours, years, etc) because this would tell us how fast is something being done, consumed, etc. Example, the speed or rate of a car is 4 meters / 5 seconds.
A businessman has 10 employees. His salary is equal to 6 times the average of employees salaries. If the 11 of them receive a total of $64,000 in one year, what was the business mans salary that year ??
Answers
x = business man's salary
y = employee's salary
x + 10y = 64000
x = 6y
6y + 10y = 64000
16y = 64000
y = 64000/16 = 4000
x = 6*4000
x = 24000
business man's salary was $24,000
One serving of granola provides 4% of the protein you need daily. You must get the remaining 48 grams from other sources. How many grams of protein do you need daily?
Answers
x grams. = 4%
Set up a proportion
48/96 = x/4
96/24=4
48/24=2
x=2= grams in the granola
Add the value of 96% and the value of 4% to get the value of 100%
2+48=50
Final answer: 50 grams
If the total price paid for an item is $20.25 and the cost of the item before sales tax is $18.90, what is the rate of sales tax?
Answers
20.25 -18.90 = 1.35
1.35 / 18.90 = 0.071428571
= 7.14%
check:
18.90 x 1.34946 Round to 1.35
18.90 + 1.35 = 20.25
Sales tax = 7.14%
Compare the 3 in 536 to the 3 in 1,36 7.
Draw number 536 using base ten block?
Answers
For the number 1367. the order of the numbers from left to right is "thousands hundreds tens units", this means that the 3 is occupying the hundreds place.
As per the representation of 536 using base 10 blocks, it is shown in the attached picture:
The scores on an exam are normally distributed, with a mean of 74 and a standard deviation of 7. What percent of the scores are less than 81?The scores on an exam are normally distributed, with a mean of 74 and a standard deviation of 7. What percent of the scores are less than 81?
Answers
We are given the following variables:
Normal distribution
u = sample mean = 74
s = standard deviation of the samples = 7
x = sample value = 81
Since we are asked to find percent of the scores less than 81, this means we have to find the proportion using the z-score. The formula for z score is:
z = (x – u) / s
Substituting the given values:
z = (81 – 74) / 7
z = 1
Using the standard normal distribution tables for z, we find that:
P = 0.8413 at z = 1
Therefore this means that 84.13% of scores are less than 81.
The Answer is the two small boxes just got it right For Plato
FAR LEFT FAR RIGHT
is a kiloliter greater than one liter
Answers
Kiloliters is a metric unit of volume equivalent to 1000 liters or 1 cube meter.
-------------------------------------------------------------------------------------------
Yes, a kiloliter is greater than one liter.
------------------------------------
1 kiloliter = 1000 liters
I hope that helps!
Find the taylor polynomial t3(x) for the function f centered at the number
a. f(x) = e^(â3x) sin(2x), a = 0
Answers
[tex]\sin2x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n(2x)^{2n+1}}{(2n+1)!}=2x-\dfrac{4x^3}3+\cdots[/tex]
[tex]e^{-3x}\sin2x=\left(1-3x+9x^2+\cdots\right)\left(2x-\dfrac{4x^3}3+\cdots\right)[/tex]
[tex]\approx T_3(x)=(1-3x+9x^2)\left(2x-\dfrac{4x^3}3\right)[/tex]
[tex]T_3(x)=2x-6x^2+\left(18-\dfrac43\right)x^3[/tex]
[tex]T_3(x)=2x-6x^2+\dfrac{50}3x^3[/tex]
Cos (pi/11)+cos (3pi/11)+cos (5pi/11)+cos (7pi/11)+cos(9pi/11)
Answers
Answer:
1/2
Step-by-step explanation:
((81x^3 y^(-1/3))^1/4)/(x^2 y)^1/4 simplify
Answers
[tex]\bf \cfrac{\left( 81x^3y^{-\frac{1}{3}} \right)^{\frac{1}{4}}}{(x^2y)^{\frac{1}{4}}}\implies \cfrac{81^{\frac{1}{4}}x^{3\cdot {\frac{1}{4}}}y^{-\frac{1}{3}\cdot {\frac{1}{4}}}}{x^{2\cdot {\frac{1}{4}}}y^{\frac{1}{4}}}\implies \cfrac{81^{\frac{1}{4}}x^{\frac{3}{4}}y^{-\frac{1}{12}}}{x^{\frac{2}{4}} y^{\frac{1}{4}}}[/tex]
[tex]\bf \cfrac{81^{\frac{1}{4}}x^{\frac{3}{4}}x^{-\frac{2}{4}}}{ y^{\frac{1}{4}}y^{\frac{1}{12}}}\implies \cfrac{(3^4)^{\frac{1}{4}}x^{\frac{3}{4}-\frac{2}{4}}}{y^{\frac{1}{4}+\frac{1}{12}}}\implies \cfrac{3^{4\cdot \frac{1}{4}}x^{\frac{1}{4}}}{y^{\frac{4}{12}}}\implies \cfrac{3^{\frac{4}{4}}x^{\frac{1}{4}}}{y^{\frac{1}{3}}} \\\\\\ \cfrac{3\sqrt[4]{x}}{\sqrt[3]{y}}[/tex]
Simplify x^2-4x-32/x^2+7x+12
Answers
(x - 8)(x + 4) / (x+3)(x+4)
If x ≠ -4
Then it equals to:
(x-8) / (x+3)
Answer see form:
how do you make a equivalent ratio?
Answers
a commercial jet airplane climbs at takeoff with slope m=4/9. How far in the horizontal direction will the airplane fly to reach am altitude of 20,000 ft above the takeoff point?
Answers
solve for "h".
The horizontal direction in will the airplane flies to reach an altitude of 20,000 ft above the takeoff point will be 45,000 feet.
What is a linear equation?A relationship between two or more parameters that, when shown on a graph, produces a linear model. The degree of the variable will be one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
A commercial jet airplane climbs at takeoff with slope m = 4/9.
Then the equation of the line will be given as,
y = (4/9)x
The y-intercept is zero.
The horizontal direction will the airplane fly to reach an altitude of 20,000 ft above the takeoff point will be
20,000 = (4/9)x
180,000 = 4x
x = 180,000 / 4
x = 45,000 feet
The horizontal direction in will the airplane flies to reach an altitude of 20,000 ft above the takeoff point will be 45,000 feet.
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Find the area of the shaded region when r = 4 feet and the area of the triangle is: 123√ square feet
Use 1.73 for 3√ and 3.14 for π. Round your answer to two decimal places.
Answers
A ( shaded ) = A ( circle ) - A ( triangle ) =
= r² π - 12 √ 3 =
= 4² π - 12 √ 3 =
= 16 π - 12 √ 3 =
= 16 · 3.14 - 12 · 1.73 =
= 50.24 - 20.76 = 29.48
Answer: Area of the shaded region is 29.48 ft².
Product A is an 8 oz bottle of cough medication that sells for $1.36. Product B is a 16oz bottle of cough medication that cost $3.20. Which product has the lower unit price
Answers
^^^^^^^^^^^^^^^^
If 4 is subtracted from twice a number, the result is 10 less than the number. Write an equation to solve this problem. Use n as the variable to represent the number.
Answers
-n. -n
n-4=-10
+4. +4
n=-6
Final answer: -6
A group of ladies meet for an afternoon tea party. They bring all their cats. In all, there are 26 heads and 84 feet. How many ladies and how many cats are in the room
Answers
X= ladies y = number of cats
Each lady has 2 feet and each cat has 4 feet
<=>
x + y = 26 (1) => number of characters
2x + 4y = 84 (2) => number of feet
<=> operation : (1) * 2
2x + 2y = 52 (1')
2x + 4y = 88 (2')
<=> operation : (2') - (1')
2y = 26
<=>
y = 20 and x = 32- 16 = 16
10 ladies and 16 cats
p+c=26, solve for c, subtract p from both sides
c=26-p, now for feet...
2p+4c=f, using c found above gives you:
2p+4(26-p)=f and we are told that there are 84 feet so:
2p+4(26-p)=84 perform indicated multiplication on left side
2p+104-4p=84, combine like terms on left side
-2p+104=84 subtract 104 from both sides
-2p=-20 divide both sides by -2
p=10, and since c=26-p
c=26-10
c=16
So there are 10 people and 16 cats in the room.
The third term of an arithmetic sequence is 21, and the eighth term is 56. The first term is _____.
Answers
The first term of the given arithmetic sequence is 7. This can be determined by finding the common difference and moving backwards from the third term using this value.
Explanation:The subject of this question is Arithmetic Sequence which is a topic in mathematics. An arithmetic sequence is defined by a common difference (d) between consecutive terms.
To solve this problem, we have to find the common difference first. The difference between the 8th term (56) and the 3rd term (21) is 35. But since there are 5 steps from the 3rd term to the 8th term, each step (or the common difference) must be 35 ÷ 5 = 7.
Now that we have the common difference (7), we can reverse it from the 3rd term (21) back to the 1st term which gives us: 21 - 2(7) = 7. So, the first term is 7.
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The volume of a rectangular prism is b3 + 8b2 + 19b + 12 cubic units, and its height is b + 3 units. The area of the base of the rectangular prism is square units.
Answers
The volume of the rectangular prism is
[tex] b^{3}+8 b^{2} +19b+12 [/tex],
the height is h=(b+3)
1. The volume of a rectangular prism is (base area)*height
also, notice that the volume is a third degree polynomial, the height is a 1st degree polynomial, so the base area must be a 2nd degree polynomial, whose coefficients we don't know yet.
Let this quadratic polynomial be [tex](mb^{2}+nb+k)[/tex]
2
[tex]b^{3}+8 b^{2} +19b+12=(mb^{2}+nb+k)*(b+3)[/tex]
notice that [tex]b^{3}[/tex] is the product of the largest 2 terms: [tex]mb^{2}[/tex] and b, so m must be 1
also, notice that 12 is the product of the constants, k and 3
so k*3=12, this means k=4
3
we write the above equality again:
[tex] b^{3}+8 b^{2} +19b+12=(b^{2}+nb+4)*(b+3)[/tex]
[tex](b^{2}+nb+4)(b+3)= b^{3}+3 b^{2} +nb^{2}+3nb+4b+12[/tex]
=[tex]= b^{3}+(n+3)b^{2}+(3n+4)b+12[/tex]
4
now compare the coefficient with the left side:
[tex]8 b^{2}=(n+3)b^{2}[/tex]
8=n+3
n=5
substituting n=5:
the base area is [tex]b^{2}+5b+4 [/tex]
Answer: [tex]b^{2}+5b+4 [/tex]
Answer:
b^2 + 5b +4
Step-by-step explanation:
Three students simplified the expression 2x- 3(y-2x) +-5times -2y.
Amber -4y + 7y
Butch 4x + 7y
Carl 8x + 7y
Who is correct? Explain the error the other students made
Answers
What is the estimate of 462,809+256,738
Answers
the estimate would be...
500,000 + 300,000
your answer would be: 800,000.
I hoped I helped and if you need more you could always ask me :)
-Dawn
The function f(x)=log[7]x is increasing. True or false
Answers
The given function increases as x increases, so the statement is true.
Is the function increasing?Here we have the function:
[tex]f(x) = log_7(x)[/tex]
To see if it is increasing, we can evaluate in two values, 10 and 15, this will serve to see if the function is increasing or not:
[tex]f(10) = log_7(10) = 1.18\\\\f(15) = log_7(15) = 1.39[/tex]
As you can see, when x increases also does f(x), so the function is increasing.
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