MATHEMATICS HIGH SCHOOL

Y varies directly as x , y= 25 when x=5. Determine y when x= 13

Answers

Answer 1

Answer:

y = 65 when x = 13

Step-by-step explanation:

Here we have a proportion problem.

Y varies directly as x means that y equals the product of x and a constant

Let’s say our constant is k

Thus;

y = kx

now, k = y/x

Using the initial values;

k = 25/5 = 5

Now we want to get y when x = 13

Recall; y = kx

Thus using the value of k earlier calculated;

y = 13 * 5

y = 65

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To aid in sea navigation, Little Gull Island Lighthouse shines a light from a height of 91 feet above sea level with an unknown angle of depression. If the beam of light shines on the sea surface at a point that is 865 feet away from the base of the lighthouse, what is the angle of depression?

Answers

Answer:

$\approx 6^\circ$

Step-by-step explanation:

Given that:

Little Gull Island Lighthouse shines a light from a height of 91 feet above the sea level.

The angle of depression is unknown.

Distance of the point at sea surface from the base of lighthouse is 865 ft.

This situation can be modeled or can be represented as the figure attached in the answer area.

The situation can be represented by a right angled $\triangle ABC$ in which we are given the base and the height of the triangle.

And we have to find the value of $\angle BAD \ or \ \angle C$ (Because they are the internal vertically opposite angles).

Using tangent ratio:

$tan\theta = (Perpendicular)/(Base)$

$tanC = (AC)/(BC)\n\Rightarrow tanC = (91)/(865)\n\Rightarrow tanC = 0.105\n\Rightarrow \angle C \approx 6^\circ$

Therefore, the angle of depression is: $\approx 6^\circ$

Write the given expression in terms of x and y only tan(sin−1 x + cos−1 y)

Answers

Answer:

$\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=(xy+√(\left ( 1-x^2 \right )\left ( 1-y^2 \right )))/(y√(1-x^2)-x√(1-y^2))$

Step-by-step explanation:

We need to express $\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )$ in terms of x and y .

Let $\sin ^(-1)x=\theta \,,\,\cos ^(-1)y=\phi$ , we get

$x=\sin \theta \,,\,y=\cos \phi$

Formulae Used:

$\sin ^2\theta +\cos ^2\theta =1\n\sin ^2 \phi +\cos ^2 \phi =1$

$\cos \theta =√(1-\sin^2 \theta)=√(1-x^2)\n\sin \phi=√(1-\cos ^2 \phi)=√(1-y^2)$

$\tan \theta =(\sin \theta )/(\cos\theta )=(x)/(√(1-x^2))\n\tan \phi =(\sin \phi)/(\cos\phi)=(√(1-y^2))/(y)$

We know that $\tan \left ( \theta +\phi\right )=(\tan \theta +\tan \phi )/(1-\tan \theta \,\tan \phi )$

$\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=((x)/(√(1-x^2))+(√(1-y^2))/(y))/(1-(x√(1-y^2))/(y√(-x^2)))\n\therefore \tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=(xy+√(\left ( 1-x^2 \right )\left ( 1-y^2 \right )))/(y√(1-x^2)-x√(1-y^2))$

What is the first derivative of r with respect to t (i.e., differentiate r with respect to t)? r = 5/(t2)Note: Use ^ to show exponents in your answer, so for example x2 = x^2. Also, type your equation answer without additional spaces.

Answers

Answer:

The first derivative of $r(t) = 5\cdot t^(-2)$ (r(t)=5*t^{-2}) with respect to t is $r'(t) = -10\cdot t^(-3)$ (r'(t) = -10*t^{-3}).

Step-by-step explanation:

Let be $r(t) = (5)/(t^(2))$ , which can be rewritten as $r(t) = 5\cdot t^(-2)$ . The rule of differentiation for a potential function multiplied by a constant is:

$(d)/(dt)(c \cdot t^(n)) = n\cdot c \cdot t^(n-1)$ , $\forall \,n\neq 0$

Then,

$r'(t) = (-2)\cdot 5\cdot t^(-3)$

$r'(t) = -10\cdot t^(-3)$ (r'(t) = -10*t^{-3})

The first derivative of $r(t) = 5\cdot t^(-2)$ (r(t)=5*t^{-2}) with respect to t is $r'(t) = -10\cdot t^(-3)$ (r'(t) = -10*t^{-3}).

80,000,000 in expanded form, word form ,standard form

Answers

80,000,000 = 80,000,000 (in expanded form)
80,000,000 = eighty million (in word form)
80,000,000 = 8 x 10^7 (in standard form)

Answer:

Answer in standard form: 80,000,000

Expanded form: 80,000,000+0000000

Word form: eighty million

Step-by-step explanation:

Round the decimal number to the nearest hundredth 5.6192

Answers

the 1 is the digit in te hundredths place, so you need to look at the digit directly to the right of that, the 9. When rounding, if the digit to the right pf the place you're rounding is 0-4, the number you're rounding stays the same, and all digits to the right become zeros. If that digit is 5-9, the digit you're rounding goes up to the next number, and all digits to the right become zeros. So, because the digit to the right of the hundredths place is a 9, that tells you that the 1 has to go up to the next number in counting order, so the number 5.6192 becomes 5.62 when rounded to the hundredths place.

A commercial aircraft gets the best fuel efficiency if it operates at a minimum altitude of 29,000 feet and a maximum altitude of 41,000 feet. Model the most fuel-efficient altitudes using a compound inequality.