Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Erika was given this problem: The side s(t) of a cube is increasing at a rate of 2 kilometers per hour. At a certain instant t_(0), the side is 1.5 kilometers. What is the rate of change of the volume V(t) of the cube at that instant? Which equation should Erika use to solve the problem? Choose 1 answer: (A) V(t)=sqrt(3[s(t)]^(2)) (B) V(t)=[s(t)]^(2) (C) V(t)=6[s(t)]^(2) (D) V(t)=[s(t)]^(3)

Erika was given this problem:\newlineThe side s(t) s(t) of a cube is increasing at a rate of 22 kilometers per hour. At a certain instant t0 t_{0} , the side is 11.55 kilometers. What is the rate of change of the volume V(t) V(t) of the cube at that instant?\newlineWhich equation should Erika use to solve the problem?\newlineChoose 11 answer:\newline(A) V(t)=3[s(t)]2 V(t)=\sqrt{3[s(t)]^{2}} \newline(B) V(t)=[s(t)]2 V(t)=[s(t)]^{2} \newline(C) V(t)=6[s(t)]2 V(t)=6[s(t)]^{2} \newline(D) V(t)=[s(t)]3 V(t)=[s(t)]^{3}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Grade 7right chevron icon
Identify equivalent linear expressions: word problemsright chevron icon

Full solution

Q. Erika was given this problem:\newlineThe side s(t) s(t) of a cube is increasing at a rate of 22 kilometers per hour. At a certain instant t0 t_{0} , the side is 11.55 kilometers. What is the rate of change of the volume V(t) V(t) of the cube at that instant?\newlineWhich equation should Erika use to solve the problem?\newlineChoose 11 answer:\newline(A) V(t)=3[s(t)]2 V(t)=\sqrt{3[s(t)]^{2}} \newline(B) V(t)=[s(t)]2 V(t)=[s(t)]^{2} \newline(C) V(t)=6[s(t)]2 V(t)=6[s(t)]^{2} \newline(D) V(t)=[s(t)]3 V(t)=[s(t)]^{3}
  1. Volume Formula: To find the rate of change of the volume of the cube, we need to use the formula for the volume of a cube, which is the side length cubed, V(t)=[s(t)]3V(t) = [s(t)]^3. This is because the volume of a cube is calculated by multiplying the length of the side by itself three times.
  2. Differentiation with Chain Rule: Since the side s(t)s(t) is changing with time, we need to differentiate the volume function V(t)V(t) with respect to time tt to find the rate of change of the volume. This will give us dVdt\frac{dV}{dt}, the rate of change of the volume with respect to time.
  3. Given Rates and Values: Differentiating V(t)=[s(t)]3V(t) = [s(t)]^3 with respect to time tt, we use the chain rule which states that dVdt=3[s(t)]2⋅dsdt\frac{dV}{dt} = 3[s(t)]^2 \cdot \frac{ds}{dt}, where dsdt\frac{ds}{dt} is the rate of change of the side length with respect to time.
  4. Substitution and Simplification: We are given that the rate of change of the side length, dsdt\frac{ds}{dt}, is 22 kilometers per hour. We also know that at the instant t0t_0, the side length s(t)s(t) is 1.51.5 kilometers.
  5. Calculation of Rate: Substituting the given values into the differentiated volume formula, we get dVdt=3[1.5]2×2\frac{dV}{dt} = 3[1.5]^2 \times 2. This simplifies to dVdt=3×2.25×2\frac{dV}{dt} = 3 \times 2.25 \times 2.
  6. Calculation of Rate: Substituting the given values into the differentiated volume formula, we get dVdt=3[1.5]2×2\frac{dV}{dt} = 3[1.5]^2 \times 2. This simplifies to dVdt=3×2.25×2\frac{dV}{dt} = 3 \times 2.25 \times 2. Calculating the value, we get dVdt=3×2.25×2=13.5 km3/hour\frac{dV}{dt} = 3 \times 2.25 \times 2 = 13.5 \text{ km}^3/\text{hour}. This is the rate of change of the volume of the cube at the instant t0t_0.

More problems from Identify equivalent linear expressions: word problems

Question
Simplify the expression:\newline7z2+3z27z^2 + 3z^2 \newline ______
Get tutor helpright-arrow

Posted 7 months ago

Question
Simplify the expression:\newline10f+8f−9f−6f10f + 8f - 9f - 6f\newline____
Get tutor helpright-arrow

Posted 7 months ago

Question
Subtract.\newline(3h+6)−(3h+2)(3h + 6) - (3h + 2)\newline____
Get tutor helpright-arrow

Posted 7 months ago

Question
Simplify the expression:\newline2p−p2p - p \newline______
Get tutor helpright-arrow

Posted 8 months ago

Question
A school charges $6\$6 per ticket to a dance. They spend $300\$300 on music and decorations. The expression 6n−3006n-300 describes the amount of profit the school makes from the dance if nn students buy tickets. What does 6n6n represent in this context?\newlineChoices:\newline(A)The amount of money the school charges for all `\n` tickets\newline(B)The amount of money the school spends on the dance\newline(C)The number of students who buy tickets\newline(D)The profit the school earns per ticket
Get tutor helpright-arrow

Posted 8 months ago

Question
Which is equal to cos⁡60∘ \cos 60^\circ ? \newlineChoices:\newline(A) sin⁡30∘ \sin 30^\circ \newline(B) sin⁡60∘ \sin 60^\circ \newline(C) cos⁡30∘ \cos 30^\circ \newline(D) cos⁡60∘ \cos 60^\circ
Get tutor helpright-arrow

Posted 9 months ago

Question
Which is equal to sin⁡45∘\sin 45^\circ? \newlineChoices:\newline(A) cos⁡45∘\cos 45^\circ\newline(B) cos⁡55∘\cos 55^\circ\newline(C) sin⁡55∘\sin 55^\circ\newline(D) sin⁡35∘\sin 35^\circ
Get tutor helpright-arrow

Posted 7 months ago

Question
Which is equal to cos⁡75∘\cos 75^\circ? \newlineChoices:\newline(A) sin⁡15∘\sin 15^\circ\newline(B) sin⁡75∘\sin 75^\circ\newline(C) cos⁡15∘\cos 15^\circ\newline(D) cos⁡25∘\cos 25^\circ
Get tutor helpright-arrow

Posted 7 months ago

Question
Which is equal to sin⁡30∘ \sin 30^\circ ? \newlineChoices:\newline(A) cos⁡60∘ \cos 60^\circ \newline(B) cos⁡30∘ \cos 30^\circ \newline(C) sin⁡60∘ \sin 60^\circ \newline(D) sin⁡30∘ \sin 30^\circ
Get tutor helpright-arrow

Posted 7 months ago

Question
Which is equal to cos⁡45∘ \cos 45^\circ ? \newlineChoices:\newline(A) sin⁡45∘ \sin 45^\circ \newline(B) sin⁡55∘ \sin 55^\circ \newline(C) cos⁡55∘ \cos 55^\circ \newline(D) cos⁡35∘ \cos 35^\circ
Get tutor helpright-arrow

Posted 7 months ago

View all (17)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant